| L(s) = 1 | − 3·4-s − 2·5-s + 11-s + 5·16-s + 6·20-s − 8·23-s + 3·25-s − 16·31-s − 4·37-s − 3·44-s + 24·47-s − 14·49-s + 4·53-s − 2·55-s − 8·59-s − 3·64-s − 32·67-s − 16·71-s − 10·80-s − 20·89-s + 24·92-s + 20·97-s − 9·100-s − 8·103-s + 12·113-s + 16·115-s + 121-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 0.894·5-s + 0.301·11-s + 5/4·16-s + 1.34·20-s − 1.66·23-s + 3/5·25-s − 2.87·31-s − 0.657·37-s − 0.452·44-s + 3.50·47-s − 2·49-s + 0.549·53-s − 0.269·55-s − 1.04·59-s − 3/8·64-s − 3.90·67-s − 1.89·71-s − 1.11·80-s − 2.11·89-s + 2.50·92-s + 2.03·97-s − 0.899·100-s − 0.788·103-s + 1.12·113-s + 1.49·115-s + 1/11·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695275 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695275 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( 1 - T \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−7.25611852136541238398716522907, −7.06776652987294055072227501878, −6.08693701335829428157980562583, −5.89992194836808220350577312387, −5.54640979880007283696239996782, −4.87806253960044967314702320131, −4.52659700416500112028212971426, −4.10497602187160877381582357369, −3.79000489498233733500749844862, −3.40349928139289108990558562724, −2.73199597836315214499399961863, −1.86919846930486895513918273600, −1.27797148118613320465803316120, 0, 0,
1.27797148118613320465803316120, 1.86919846930486895513918273600, 2.73199597836315214499399961863, 3.40349928139289108990558562724, 3.79000489498233733500749844862, 4.10497602187160877381582357369, 4.52659700416500112028212971426, 4.87806253960044967314702320131, 5.54640979880007283696239996782, 5.89992194836808220350577312387, 6.08693701335829428157980562583, 7.06776652987294055072227501878, 7.25611852136541238398716522907
