| L(s) = 1 | + 2·2-s − 4-s − 8·8-s − 3·9-s − 11-s − 7·16-s + 12·17-s − 6·18-s − 2·22-s + 25-s + 12·29-s − 16·31-s + 14·32-s + 24·34-s + 3·36-s − 4·37-s + 4·41-s + 44-s − 14·49-s + 2·50-s + 24·58-s − 32·62-s + 35·64-s − 32·67-s − 12·68-s + 24·72-s − 8·74-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s − 2.82·8-s − 9-s − 0.301·11-s − 7/4·16-s + 2.91·17-s − 1.41·18-s − 0.426·22-s + 1/5·25-s + 2.22·29-s − 2.87·31-s + 2.47·32-s + 4.11·34-s + 1/2·36-s − 0.657·37-s + 0.624·41-s + 0.150·44-s − 2·49-s + 0.282·50-s + 3.15·58-s − 4.06·62-s + 35/8·64-s − 3.90·67-s − 1.45·68-s + 2.82·72-s − 0.929·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 299475 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 299475 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−8.551228798296751854049846045093, −8.248642768475844318802620430882, −7.65244290168268394743634881542, −7.25611852136541238398716522907, −6.16821463973303244938343944630, −6.08693701335829428157980562583, −5.54640979880007283696239996782, −5.09766059609940450216711100862, −4.87806253960044967314702320131, −4.10497602187160877381582357369, −3.40349928139289108990558562724, −3.25423179226253980082861279896, −2.73199597836315214499399961863, −1.27797148118613320465803316120, 0,
1.27797148118613320465803316120, 2.73199597836315214499399961863, 3.25423179226253980082861279896, 3.40349928139289108990558562724, 4.10497602187160877381582357369, 4.87806253960044967314702320131, 5.09766059609940450216711100862, 5.54640979880007283696239996782, 6.08693701335829428157980562583, 6.16821463973303244938343944630, 7.25611852136541238398716522907, 7.65244290168268394743634881542, 8.248642768475844318802620430882, 8.551228798296751854049846045093
