| L(s) = 1 | − 10·3-s + 7·4-s + 45·9-s − 70·12-s + 23·16-s + 6·17-s + 4·23-s − 3·25-s − 112·27-s − 12·29-s + 315·36-s + 46·43-s − 230·48-s + 29·49-s − 60·51-s − 4·53-s − 34·61-s + 42·64-s + 42·68-s − 40·69-s + 30·75-s − 74·79-s + 140·81-s + 120·87-s + 28·92-s − 21·100-s + 14·101-s + ⋯ |
| L(s) = 1 | − 5.77·3-s + 7/2·4-s + 15·9-s − 20.2·12-s + 23/4·16-s + 1.45·17-s + 0.834·23-s − 3/5·25-s − 21.5·27-s − 2.22·29-s + 52.5·36-s + 7.01·43-s − 33.1·48-s + 29/7·49-s − 8.40·51-s − 0.549·53-s − 4.35·61-s + 21/4·64-s + 5.09·68-s − 4.81·69-s + 3.46·75-s − 8.32·79-s + 15.5·81-s + 12.8·87-s + 2.91·92-s − 2.09·100-s + 1.39·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.190294928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.190294928\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( ( 1 + T^{2} )^{3} \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 7 T^{2} + 13 p T^{4} - 63 T^{6} + 13 p^{3} T^{8} - 7 p^{4} T^{10} + p^{6} T^{12} \) |
| 3 | \( ( 1 + 5 T + 5 p T^{2} + 31 T^{3} + 5 p^{2} T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 7 | \( 1 - 29 T^{2} + 397 T^{4} - 3401 T^{6} + 397 p^{2} T^{8} - 29 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 49 T^{2} + 1077 T^{4} - 14497 T^{6} + 1077 p^{2} T^{8} - 49 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 - 3 T + 33 T^{2} - 75 T^{3} + 33 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( 1 - 32 T^{2} - 272 T^{4} + 20563 T^{6} - 272 p^{2} T^{8} - 32 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 - 2 T + 12 T^{2} - 21 T^{3} + 12 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 + 6 T + 36 T^{2} + 41 T^{3} + 36 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 85 T^{2} + 4001 T^{4} - 144273 T^{6} + 4001 p^{2} T^{8} - 85 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( 1 - 195 T^{2} + 16670 T^{4} - 800807 T^{6} + 16670 p^{2} T^{8} - 195 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 - 52 T^{2} + 2288 T^{4} - 148785 T^{6} + 2288 p^{2} T^{8} - 52 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 - 23 T + 303 T^{2} - 2411 T^{3} + 303 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 136 T^{2} + 7400 T^{4} - 300717 T^{6} + 7400 p^{2} T^{8} - 136 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 + 2 T + 158 T^{2} + 211 T^{3} + 158 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 5 p T^{2} + 38966 T^{4} - 2954787 T^{6} + 38966 p^{2} T^{8} - 5 p^{5} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 + 17 T + 277 T^{2} + 2243 T^{3} + 277 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 159 T^{2} + 12822 T^{4} - 768971 T^{6} + 12822 p^{2} T^{8} - 159 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( 1 - 197 T^{2} + 27105 T^{4} - 2217377 T^{6} + 27105 p^{2} T^{8} - 197 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( 1 + 2 T^{2} + 8111 T^{4} + 113148 T^{6} + 8111 p^{2} T^{8} + 2 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 37 T + 649 T^{2} + 7063 T^{3} + 649 p T^{4} + 37 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 129 T^{2} + 2477 T^{4} + 567175 T^{6} + 2477 p^{2} T^{8} - 129 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 - 139 T^{2} + 12246 T^{4} - 1103599 T^{6} + 12246 p^{2} T^{8} - 139 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 - 436 T^{2} + 88832 T^{4} - 10821345 T^{6} + 88832 p^{2} T^{8} - 436 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.70418290650293199729120014120, −5.67438786895648331391453968479, −5.51253813142228832628417221426, −5.04216256239109933628497887838, −4.81527995043297624951787616232, −4.68022347047030210775763927277, −4.56448678163307908050769031652, −4.54603385692484443699563048003, −4.21812036314208985947013128664, −3.95107095614143619059865110303, −3.93937639531506961987706567725, −3.35890545463416287192350696956, −3.25504997469468640789351320769, −3.09155909457486913056343393589, −2.84568155050695396263667364659, −2.73134804872679795630570967802, −2.50426655578701489402711116836, −2.02906061401676048334999734292, −2.02431824320083092752113002932, −1.83169866127678943973705320566, −1.48101973210686615876398255151, −1.07493251879020677683581474790, −0.74619656443806649656025081230, −0.60884228691609025056961329716, −0.48368720322470370609101972656,
0.48368720322470370609101972656, 0.60884228691609025056961329716, 0.74619656443806649656025081230, 1.07493251879020677683581474790, 1.48101973210686615876398255151, 1.83169866127678943973705320566, 2.02431824320083092752113002932, 2.02906061401676048334999734292, 2.50426655578701489402711116836, 2.73134804872679795630570967802, 2.84568155050695396263667364659, 3.09155909457486913056343393589, 3.25504997469468640789351320769, 3.35890545463416287192350696956, 3.93937639531506961987706567725, 3.95107095614143619059865110303, 4.21812036314208985947013128664, 4.54603385692484443699563048003, 4.56448678163307908050769031652, 4.68022347047030210775763927277, 4.81527995043297624951787616232, 5.04216256239109933628497887838, 5.51253813142228832628417221426, 5.67438786895648331391453968479, 5.70418290650293199729120014120
Plot not available for L-functions of degree greater than 10.
