L(s) = 1 | − 3·4-s + 5-s − 6·9-s − 2·11-s + 5·16-s − 8·19-s − 3·20-s + 25-s + 12·29-s − 16·31-s + 18·36-s + 4·41-s + 6·44-s − 6·45-s − 14·49-s − 2·55-s + 8·59-s − 20·61-s − 3·64-s + 16·71-s + 24·76-s + 16·79-s + 5·80-s + 27·81-s + 20·89-s − 8·95-s + 12·99-s + ⋯ |
L(s) = 1 | − 3/2·4-s + 0.447·5-s − 2·9-s − 0.603·11-s + 5/4·16-s − 1.83·19-s − 0.670·20-s + 1/5·25-s + 2.22·29-s − 2.87·31-s + 3·36-s + 0.624·41-s + 0.904·44-s − 0.894·45-s − 2·49-s − 0.269·55-s + 1.04·59-s − 2.56·61-s − 3/8·64-s + 1.89·71-s + 2.75·76-s + 1.80·79-s + 0.559·80-s + 3·81-s + 2.11·89-s − 0.820·95-s + 1.20·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15125 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$ | \( 1 - T \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−10.82193865307292862841462120688, −10.44930765910572775087302550356, −9.365267708430525132762162914792, −9.336297212975655357629362917616, −8.631003161371790257232477095324, −8.248642768475844318802620430882, −7.82885193609770565758085877155, −6.54918508631964327843106616026, −6.16821463973303244938343944630, −5.23353820210759730524176244525, −5.09766059609940450216711100862, −4.11816586584967359873614255585, −3.25423179226253980082861279896, −2.31804956482818418780108048277, 0,
2.31804956482818418780108048277, 3.25423179226253980082861279896, 4.11816586584967359873614255585, 5.09766059609940450216711100862, 5.23353820210759730524176244525, 6.16821463973303244938343944630, 6.54918508631964327843106616026, 7.82885193609770565758085877155, 8.248642768475844318802620430882, 8.631003161371790257232477095324, 9.336297212975655357629362917616, 9.365267708430525132762162914792, 10.44930765910572775087302550356, 10.82193865307292862841462120688