| L(s) = 1 | + 2·5-s − 2·7-s − 2·11-s + 6·13-s − 4·17-s + 6·23-s + 3·25-s + 10·29-s + 4·31-s − 4·35-s + 4·37-s − 2·41-s + 6·43-s + 4·47-s + 2·49-s − 4·53-s − 4·55-s + 10·59-s + 6·61-s + 12·65-s + 16·67-s − 22·71-s + 18·73-s + 4·77-s + 16·79-s − 6·83-s − 8·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s − 0.603·11-s + 1.66·13-s − 0.970·17-s + 1.25·23-s + 3/5·25-s + 1.85·29-s + 0.718·31-s − 0.676·35-s + 0.657·37-s − 0.312·41-s + 0.914·43-s + 0.583·47-s + 2/7·49-s − 0.549·53-s − 0.539·55-s + 1.30·59-s + 0.768·61-s + 1.48·65-s + 1.95·67-s − 2.61·71-s + 2.10·73-s + 0.455·77-s + 1.80·79-s − 0.658·83-s − 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4665600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4665600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.399807349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.399807349\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 70 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 53 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 82 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 97 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 130 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 79 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 16 T + 146 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 22 T + 250 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 18 T + 214 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16 T + 209 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 70 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.235289449896469766776963861573, −8.837742923231340671809744308312, −8.539332772829641789898956028424, −8.333270517612557651835064641385, −7.73693540045182477010580738940, −7.26668534204092141823126422625, −6.64235463329142109884693225609, −6.59420086222669993869529697649, −6.20999864237535001196941073284, −5.85374606071339145288905515385, −5.14080040216122721109849735810, −5.12387840669254769349257298457, −4.30659177047953783058588119882, −4.09592760996266130688407018215, −3.18705466573447478455212584926, −3.17082536459215421610774302292, −2.36538785070912119171089935931, −2.13906968512772871672474254369, −1.04750154385184985082253610685, −0.798840594505406877991127967201,
0.798840594505406877991127967201, 1.04750154385184985082253610685, 2.13906968512772871672474254369, 2.36538785070912119171089935931, 3.17082536459215421610774302292, 3.18705466573447478455212584926, 4.09592760996266130688407018215, 4.30659177047953783058588119882, 5.12387840669254769349257298457, 5.14080040216122721109849735810, 5.85374606071339145288905515385, 6.20999864237535001196941073284, 6.59420086222669993869529697649, 6.64235463329142109884693225609, 7.26668534204092141823126422625, 7.73693540045182477010580738940, 8.333270517612557651835064641385, 8.539332772829641789898956028424, 8.837742923231340671809744308312, 9.235289449896469766776963861573
