L(s) = 1 | + 4·2-s − 26.8·3-s + 16·4-s + 25·5-s − 107.·6-s − 148.·7-s + 64·8-s + 479.·9-s + 100·10-s + 389.·11-s − 430.·12-s + 170.·13-s − 594.·14-s − 672.·15-s + 256·16-s − 275.·17-s + 1.91e3·18-s + 543.·19-s + 400·20-s + 3.99e3·21-s + 1.55e3·22-s − 529·23-s − 1.72e3·24-s + 625·25-s + 683.·26-s − 6.36e3·27-s − 2.37e3·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.72·3-s + 0.5·4-s + 0.447·5-s − 1.21·6-s − 1.14·7-s + 0.353·8-s + 1.97·9-s + 0.316·10-s + 0.970·11-s − 0.862·12-s + 0.280·13-s − 0.811·14-s − 0.771·15-s + 0.250·16-s − 0.231·17-s + 1.39·18-s + 0.345·19-s + 0.223·20-s + 1.97·21-s + 0.686·22-s − 0.208·23-s − 0.609·24-s + 0.200·25-s + 0.198·26-s − 1.67·27-s − 0.573·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 4T \) |
| 5 | \( 1 - 25T \) |
| 23 | \( 1 + 529T \) |
good | 3 | \( 1 + 26.8T + 243T^{2} \) |
| 7 | \( 1 + 148.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 389.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 170.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 275.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 543.T + 2.47e6T^{2} \) |
| 29 | \( 1 + 2.84e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.60e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.34e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.23e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.94e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.59e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.18e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.60e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.10e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 9.60e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.84e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.68e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.99e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.46e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.05e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.14e5T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.10514626599153949862552881235, −10.15514977465976744699285886003, −9.259790497380393931796119843313, −7.20066353162666761944767836746, −6.33907244914933624976116072215, −5.86376162766207066881784565267, −4.70182674468700034858858486470, −3.47736711193044966067206907393, −1.49293107319338280671630371447, 0,
1.49293107319338280671630371447, 3.47736711193044966067206907393, 4.70182674468700034858858486470, 5.86376162766207066881784565267, 6.33907244914933624976116072215, 7.20066353162666761944767836746, 9.259790497380393931796119843313, 10.15514977465976744699285886003, 11.10514626599153949862552881235