| L(s) = 1 | − 4·3-s − 5·5-s + 27·9-s + 60·11-s + 172·13-s + 20·15-s − 18·17-s − 44·19-s − 48·23-s − 260·27-s − 372·29-s − 176·31-s − 240·33-s − 254·37-s − 688·39-s + 372·41-s − 200·43-s − 135·45-s − 168·47-s + 72·51-s + 498·53-s − 300·55-s + 176·57-s + 252·59-s + 58·61-s − 860·65-s + 1.03e3·67-s + ⋯ |
| L(s) = 1 | − 0.769·3-s − 0.447·5-s + 9-s + 1.64·11-s + 3.66·13-s + 0.344·15-s − 0.256·17-s − 0.531·19-s − 0.435·23-s − 1.85·27-s − 2.38·29-s − 1.01·31-s − 1.26·33-s − 1.12·37-s − 2.82·39-s + 1.41·41-s − 0.709·43-s − 0.447·45-s − 0.521·47-s + 0.197·51-s + 1.29·53-s − 0.735·55-s + 0.408·57-s + 0.556·59-s + 0.121·61-s − 1.64·65-s + 1.88·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.628890889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.628890889\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T - 11 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 60 T + 2269 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 86 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 18 T - 4589 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 44 T - 4923 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 48 T - 9863 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 186 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 176 T + 1185 T^{2} + 176 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 254 T + 13863 T^{2} + 254 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 186 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 100 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 168 T - 75599 T^{2} + 168 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 498 T + 99127 T^{2} - 498 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 252 T - 141875 T^{2} - 252 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 58 T - 223617 T^{2} - 58 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 1036 T + 772533 T^{2} - 1036 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 168 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 506 T - 132981 T^{2} + 506 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 272 T - 419055 T^{2} + 272 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 948 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 1014 T + 323227 T^{2} - 1014 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 766 T + p^{3} T^{2} )^{2} \) |
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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
−9.680192524692693922060174977826, −9.351280473214916251730519348744, −8.984284139791594729985479998617, −8.730663756636450232920417146710, −8.090058629988002058442205124043, −7.86052112490459700123453775957, −7.10593167755565717856843069433, −6.78011646536990323650252143422, −6.42593148965861970019521527090, −5.92210065467290238793167633222, −5.71775391053931475835631781916, −5.20189528290585615028239166265, −4.21256024586068209407477961747, −3.94822270483178887264705563714, −3.62274240460267753672780415235, −3.57467630122654765374657911941, −1.95840338506408931106824861124, −1.72173804728827418378758014921, −1.11648815660509852004973509023, −0.49359664596752228827508268469,
0.49359664596752228827508268469, 1.11648815660509852004973509023, 1.72173804728827418378758014921, 1.95840338506408931106824861124, 3.57467630122654765374657911941, 3.62274240460267753672780415235, 3.94822270483178887264705563714, 4.21256024586068209407477961747, 5.20189528290585615028239166265, 5.71775391053931475835631781916, 5.92210065467290238793167633222, 6.42593148965861970019521527090, 6.78011646536990323650252143422, 7.10593167755565717856843069433, 7.86052112490459700123453775957, 8.090058629988002058442205124043, 8.730663756636450232920417146710, 8.984284139791594729985479998617, 9.351280473214916251730519348744, 9.680192524692693922060174977826
