| L(s) = 1 | + 16·11-s − 72·17-s − 8·19-s + 16·25-s + 40·41-s + 80·43-s − 14·49-s + 184·59-s + 224·67-s + 232·73-s + 88·83-s − 312·89-s − 136·97-s − 96·107-s + 304·113-s − 180·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 592·169-s + 173-s + ⋯ |
| L(s) = 1 | + 1.45·11-s − 4.23·17-s − 0.421·19-s + 0.639·25-s + 0.975·41-s + 1.86·43-s − 2/7·49-s + 3.11·59-s + 3.34·67-s + 3.17·73-s + 1.06·83-s − 3.50·89-s − 1.40·97-s − 0.897·107-s + 2.69·113-s − 1.48·121-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 + 3.50·169-s + 0.00578·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.100479064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.100479064\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 16 T^{2} - 254 T^{4} - 16 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 8 T + 186 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 592 T^{2} + 143170 T^{4} - 592 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 36 T + 870 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 4 T + 4 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 268 T^{2} + 477286 T^{4} - 268 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 1178 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 1380 T^{2} + 1419974 T^{4} - 1380 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 1780 T^{2} + 2031622 T^{4} - 1780 p^{4} T^{6} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 20 T + 3174 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 40 T + 4090 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 7492 T^{2} + 23390470 T^{4} - 7492 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 1604 T^{2} - 6155034 T^{4} - 1604 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 92 T + 8836 T^{2} - 92 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 13232 T^{2} + 71110338 T^{4} - 13232 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 112 T + 11602 T^{2} - 112 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 9412 T^{2} + 47279686 T^{4} - 9412 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 116 T + 13894 T^{2} - 116 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 16900 T^{2} + 142880134 T^{4} - 16900 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 44 T + 13924 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 156 T + 20774 T^{2} + 156 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 68 T + 3046 T^{2} + 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.57065785766242096724336441415, −6.16351660299035230804707268157, −5.96921217836367218054909379179, −5.66896901177805412886437250745, −5.47523711090579121812834759080, −5.16023828800970812742764052520, −5.06637021885440238671822895099, −4.88833389682195114279930006533, −4.38585497325716329249657817517, −4.24926564155713104994021166025, −4.21695805374762711099339458163, −3.93546265980307474265407173445, −3.90137888536371055315970239886, −3.64162190836668500940307010249, −3.09288602277313798291250248819, −2.89554814856293800843005706157, −2.55967722697070426605631371167, −2.31013698089092948857975947442, −2.26741916803716910060841563111, −1.86761538093128545670378044296, −1.79278876231269490385725203916, −1.18077856323363786701161796721, −0.75524526519699635039338133128, −0.69154894167603822063880637661, −0.28273704292989391192593916218,
0.28273704292989391192593916218, 0.69154894167603822063880637661, 0.75524526519699635039338133128, 1.18077856323363786701161796721, 1.79278876231269490385725203916, 1.86761538093128545670378044296, 2.26741916803716910060841563111, 2.31013698089092948857975947442, 2.55967722697070426605631371167, 2.89554814856293800843005706157, 3.09288602277313798291250248819, 3.64162190836668500940307010249, 3.90137888536371055315970239886, 3.93546265980307474265407173445, 4.21695805374762711099339458163, 4.24926564155713104994021166025, 4.38585497325716329249657817517, 4.88833389682195114279930006533, 5.06637021885440238671822895099, 5.16023828800970812742764052520, 5.47523711090579121812834759080, 5.66896901177805412886437250745, 5.96921217836367218054909379179, 6.16351660299035230804707268157, 6.57065785766242096724336441415
