| L(s) = 1 | + 2·3-s − 2·4-s − 5-s − 3·9-s − 4·12-s + 10·13-s − 2·15-s + 4·16-s + 2·20-s + 25-s − 14·27-s − 8·31-s + 6·36-s + 4·37-s + 20·39-s − 24·41-s − 20·43-s + 3·45-s + 8·48-s + 49-s − 20·52-s + 24·53-s + 4·60-s − 8·64-s − 10·65-s − 8·67-s + 2·75-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s − 0.447·5-s − 9-s − 1.15·12-s + 2.77·13-s − 0.516·15-s + 16-s + 0.447·20-s + 1/5·25-s − 2.69·27-s − 1.43·31-s + 36-s + 0.657·37-s + 3.20·39-s − 3.74·41-s − 3.04·43-s + 0.447·45-s + 1.15·48-s + 1/7·49-s − 2.77·52-s + 3.29·53-s + 0.516·60-s − 64-s − 1.24·65-s − 0.977·67-s + 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−8.638166323935753490050543355675, −8.125489021094778603513666338176, −7.980947296320422939728903060714, −7.06609317826578845253832982017, −6.62728333206480767174228282771, −5.99075217625251652105504993885, −5.47907564040983295563221596641, −5.24423827015033096666712849344, −4.34218560100595871040786207295, −3.61689003045514298236484607491, −3.48713632587293085394972976668, −3.23792652390087211858302866834, −2.10609090528649839647621870853, −1.33767722057503568359926847519, 0,
1.33767722057503568359926847519, 2.10609090528649839647621870853, 3.23792652390087211858302866834, 3.48713632587293085394972976668, 3.61689003045514298236484607491, 4.34218560100595871040786207295, 5.24423827015033096666712849344, 5.47907564040983295563221596641, 5.99075217625251652105504993885, 6.62728333206480767174228282771, 7.06609317826578845253832982017, 7.980947296320422939728903060714, 8.125489021094778603513666338176, 8.638166323935753490050543355675
