| L(s) = 1 | − 12·25-s − 96·31-s + 372·49-s + 120·61-s + 384·79-s − 624·109-s + 420·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.64e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 0.479·25-s − 3.09·31-s + 7.59·49-s + 1.96·61-s + 4.86·79-s − 5.72·109-s + 3.47·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 9.72·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{24} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{24} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6730154382\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6730154382\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 12 T^{2} - 117 p T^{4} - 552 p^{2} T^{6} - 117 p^{5} T^{8} + 12 p^{8} T^{10} + p^{12} T^{12} \) |
| good | 7 | \( ( 1 - 186 T^{2} + 2469 p T^{4} - 1034260 T^{6} + 2469 p^{5} T^{8} - 186 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 11 | \( ( 1 - 210 T^{2} + 27603 T^{4} - 2635620 T^{6} + 27603 p^{4} T^{8} - 210 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 13 | \( ( 1 - 822 T^{2} + 306291 T^{4} - 66140428 T^{6} + 306291 p^{4} T^{8} - 822 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 17 | \( ( 1 + 228 T^{2} + 196743 T^{4} + 39926216 T^{6} + 196743 p^{4} T^{8} + 228 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 19 | \( ( 1 + 555 T^{2} - 1408 T^{3} + 555 p^{2} T^{4} + p^{6} T^{6} )^{4} \) |
| 23 | \( ( 1 + 1230 T^{2} + 1093023 T^{4} + 673333860 T^{6} + 1093023 p^{4} T^{8} + 1230 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 29 | \( ( 1 - 2004 T^{2} + 2672343 T^{4} - 2403577128 T^{6} + 2672343 p^{4} T^{8} - 2004 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 31 | \( ( 1 + 24 T + 1623 T^{2} + 42240 T^{3} + 1623 p^{2} T^{4} + 24 p^{4} T^{5} + p^{6} T^{6} )^{4} \) |
| 37 | \( ( 1 - 1338 T^{2} + 1645299 T^{4} - 3387612244 T^{6} + 1645299 p^{4} T^{8} - 1338 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 41 | \( ( 1 - 6624 T^{2} + 22038627 T^{4} - 45674785728 T^{6} + 22038627 p^{4} T^{8} - 6624 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 43 | \( ( 1 - 6630 T^{2} + 22418655 T^{4} - 49246152212 T^{6} + 22418655 p^{4} T^{8} - 6630 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 47 | \( ( 1 + 2406 T^{2} + 2511183 T^{4} - 7393099948 T^{6} + 2511183 p^{4} T^{8} + 2406 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 53 | \( ( 1 + 14052 T^{2} + 88870743 T^{4} + 321522798408 T^{6} + 88870743 p^{4} T^{8} + 14052 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 59 | \( ( 1 - 17490 T^{2} + 135320403 T^{4} - 602578996580 T^{6} + 135320403 p^{4} T^{8} - 17490 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 61 | \( ( 1 - 30 T + 5655 T^{2} - 305572 T^{3} + 5655 p^{2} T^{4} - 30 p^{4} T^{5} + p^{6} T^{6} )^{4} \) |
| 67 | \( ( 1 - 7350 T^{2} + 33720255 T^{4} - 37560148 p^{2} T^{6} + 33720255 p^{4} T^{8} - 7350 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 71 | \( ( 1 - 22854 T^{2} + 233981487 T^{4} - 1453585681428 T^{6} + 233981487 p^{4} T^{8} - 22854 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 73 | \( ( 1 - 17970 T^{2} + 190924191 T^{4} - 1223171753116 T^{6} + 190924191 p^{4} T^{8} - 17970 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 79 | \( ( 1 - 96 T + 9783 T^{2} - 339952 T^{3} + 9783 p^{2} T^{4} - 96 p^{4} T^{5} + p^{6} T^{6} )^{4} \) |
| 83 | \( ( 1 + 31350 T^{2} + 462928383 T^{4} + 4047965027700 T^{6} + 462928383 p^{4} T^{8} + 31350 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 89 | \( ( 1 - 21696 T^{2} + 318520995 T^{4} - 2898573818240 T^{6} + 318520995 p^{4} T^{8} - 21696 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 97 | \( ( 1 - 5586 T^{2} + 240370623 T^{4} - 1009488971740 T^{6} + 240370623 p^{4} T^{8} - 5586 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.43491575530694987132562789205, −2.40153506410725419901450203018, −2.21805830395748519333947020972, −2.18410016747068555969868018863, −2.12799379038620351568158773261, −2.08461527297981342389360403175, −2.03862623638183304474138076600, −1.90700110048542993510354800942, −1.80536234430551419062875105240, −1.73032074335005911753895559805, −1.46536663697368470343503353850, −1.45619677857866080137430671349, −1.43231597620540422450901059329, −1.24474962277624270980221024793, −1.07883609181809832222136913625, −0.963031518394380889715321180119, −0.899564111212257980314652330563, −0.864046394520602081152285508694, −0.849825752766200761836931143306, −0.78873323831976852921883507827, −0.68870896434340801817810623065, −0.25885079374259229655409255150, −0.24443734331334408556357795339, −0.12102342636055466454621905970, −0.07251234606325985786586395303,
0.07251234606325985786586395303, 0.12102342636055466454621905970, 0.24443734331334408556357795339, 0.25885079374259229655409255150, 0.68870896434340801817810623065, 0.78873323831976852921883507827, 0.849825752766200761836931143306, 0.864046394520602081152285508694, 0.899564111212257980314652330563, 0.963031518394380889715321180119, 1.07883609181809832222136913625, 1.24474962277624270980221024793, 1.43231597620540422450901059329, 1.45619677857866080137430671349, 1.46536663697368470343503353850, 1.73032074335005911753895559805, 1.80536234430551419062875105240, 1.90700110048542993510354800942, 2.03862623638183304474138076600, 2.08461527297981342389360403175, 2.12799379038620351568158773261, 2.18410016747068555969868018863, 2.21805830395748519333947020972, 2.40153506410725419901450203018, 2.43491575530694987132562789205
Plot not available for L-functions of degree greater than 10.
