| L(s) = 1 | + 3-s − 2·5-s + 4·11-s + 4·13-s − 2·15-s + 2·17-s + 4·19-s − 8·23-s + 5·25-s − 27-s + 12·29-s − 8·31-s + 4·33-s − 6·37-s + 4·39-s + 12·41-s − 8·43-s + 2·51-s + 2·53-s − 8·55-s + 4·57-s − 4·59-s − 2·61-s − 8·65-s − 4·67-s − 8·69-s − 16·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1.20·11-s + 1.10·13-s − 0.516·15-s + 0.485·17-s + 0.917·19-s − 1.66·23-s + 25-s − 0.192·27-s + 2.22·29-s − 1.43·31-s + 0.696·33-s − 0.986·37-s + 0.640·39-s + 1.87·41-s − 1.21·43-s + 0.280·51-s + 0.274·53-s − 1.07·55-s + 0.529·57-s − 0.520·59-s − 0.256·61-s − 0.992·65-s − 0.488·67-s − 0.963·69-s − 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.657462707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.657462707\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−9.450089286200852409431001625586, −8.550507152483425934754051193297, −8.504551369367385211718407990303, −8.188078032661879302180944799750, −7.66403392034007199236951213633, −7.21648084118425622928209066022, −7.08625306741529305752404752371, −6.34364180046167945218035585768, −6.23470169547653679603146887279, −5.68151763230287155213017980195, −5.27315478908970611821742714149, −4.60967232176112934041742650917, −4.24968154057555195751315620520, −3.76257268075429955880624032348, −3.64207021248829800110982891933, −2.90921674254738151029893545694, −2.74419785092634319081542966376, −1.52673126472786782996170552848, −1.52133213432043857164100598109, −0.56446642265163695687474452757,
0.56446642265163695687474452757, 1.52133213432043857164100598109, 1.52673126472786782996170552848, 2.74419785092634319081542966376, 2.90921674254738151029893545694, 3.64207021248829800110982891933, 3.76257268075429955880624032348, 4.24968154057555195751315620520, 4.60967232176112934041742650917, 5.27315478908970611821742714149, 5.68151763230287155213017980195, 6.23470169547653679603146887279, 6.34364180046167945218035585768, 7.08625306741529305752404752371, 7.21648084118425622928209066022, 7.66403392034007199236951213633, 8.188078032661879302180944799750, 8.504551369367385211718407990303, 8.550507152483425934754051193297, 9.450089286200852409431001625586
