| L(s) = 1 | + 4-s + 3·5-s + 9-s − 3·16-s + 3·20-s + 4·25-s − 8·31-s + 36-s + 3·45-s + 2·49-s − 12·59-s − 7·64-s − 12·71-s − 9·80-s + 81-s − 18·89-s + 4·100-s − 8·124-s − 3·125-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + 149-s + 151-s − 24·155-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.34·5-s + 1/3·9-s − 3/4·16-s + 0.670·20-s + 4/5·25-s − 1.43·31-s + 1/6·36-s + 0.447·45-s + 2/7·49-s − 1.56·59-s − 7/8·64-s − 1.42·71-s − 1.00·80-s + 1/9·81-s − 1.90·89-s + 2/5·100-s − 0.718·124-s − 0.268·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/4·144-s + 0.0819·149-s + 0.0813·151-s − 1.92·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 79 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−7.23050463680726890079276924138, −6.85675014000923957999009534498, −6.51100592762843600695651946753, −5.98770498860470517641236611057, −5.74307749942407455731996281143, −5.34468569053018482401502578444, −4.75409093070917884517149339485, −4.40401641897906227937509939922, −3.83312406758686102180441062835, −3.20103971801776995653219856016, −2.71935125721887138945505566949, −2.18682704364323426590934516962, −1.73382695584356415905826384825, −1.26409275326331310959943166221, 0,
1.26409275326331310959943166221, 1.73382695584356415905826384825, 2.18682704364323426590934516962, 2.71935125721887138945505566949, 3.20103971801776995653219856016, 3.83312406758686102180441062835, 4.40401641897906227937509939922, 4.75409093070917884517149339485, 5.34468569053018482401502578444, 5.74307749942407455731996281143, 5.98770498860470517641236611057, 6.51100592762843600695651946753, 6.85675014000923957999009534498, 7.23050463680726890079276924138
