| L(s) = 1 | + 2·2-s + 2·4-s − 8·7-s − 16·14-s − 4·16-s + 12·17-s + 16·23-s − 25-s − 16·28-s − 4·31-s − 8·32-s + 24·34-s + 8·41-s + 32·46-s − 8·47-s + 34·49-s − 2·50-s − 8·62-s − 8·64-s + 24·68-s + 16·71-s + 20·73-s − 4·79-s + 16·82-s − 8·89-s + 32·92-s − 16·94-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 3.02·7-s − 4.27·14-s − 16-s + 2.91·17-s + 3.33·23-s − 1/5·25-s − 3.02·28-s − 0.718·31-s − 1.41·32-s + 4.11·34-s + 1.24·41-s + 4.71·46-s − 1.16·47-s + 34/7·49-s − 0.282·50-s − 1.01·62-s − 64-s + 2.91·68-s + 1.89·71-s + 2.34·73-s − 0.450·79-s + 1.76·82-s − 0.847·89-s + 3.33·92-s − 1.65·94-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.347065513\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.347065513\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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
−12.03044852355822776971956849523, −11.08167584125155561638765995531, −11.07695845840639969512774657759, −10.21898822244366076901623566281, −9.632688460141725728316925891017, −9.609472923568315897006149497409, −9.176200039481152561401293759703, −8.492225997074432379526711562821, −7.63696345863375048975036301236, −7.09247381291939063699403692344, −6.80932736559196510284343795770, −6.26455237169498505723711878876, −5.82301876870837186163342380281, −5.32179651590513290569994241952, −4.90918623907008113969976221393, −3.71954981721229800996482353866, −3.53986017831255650127338570475, −3.06198098949176090838714246157, −2.70689072658095983430423910971, −0.865342802778594093176812575350,
0.865342802778594093176812575350, 2.70689072658095983430423910971, 3.06198098949176090838714246157, 3.53986017831255650127338570475, 3.71954981721229800996482353866, 4.90918623907008113969976221393, 5.32179651590513290569994241952, 5.82301876870837186163342380281, 6.26455237169498505723711878876, 6.80932736559196510284343795770, 7.09247381291939063699403692344, 7.63696345863375048975036301236, 8.492225997074432379526711562821, 9.176200039481152561401293759703, 9.609472923568315897006149497409, 9.632688460141725728316925891017, 10.21898822244366076901623566281, 11.07695845840639969512774657759, 11.08167584125155561638765995531, 12.03044852355822776971956849523
