L(s) = 1 | + 2-s + 4-s + 5-s + 4.82·7-s + 8-s + 10-s + 2·11-s − 2.82·13-s + 4.82·14-s + 16-s + 4.82·17-s + 20-s + 2·22-s + 23-s + 25-s − 2.82·26-s + 4.82·28-s − 0.828·29-s − 9.65·31-s + 32-s + 4.82·34-s + 4.82·35-s − 6.82·37-s + 40-s + 5.65·41-s − 8.48·43-s + 2·44-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s + 1.82·7-s + 0.353·8-s + 0.316·10-s + 0.603·11-s − 0.784·13-s + 1.29·14-s + 0.250·16-s + 1.17·17-s + 0.223·20-s + 0.426·22-s + 0.208·23-s + 0.200·25-s − 0.554·26-s + 0.912·28-s − 0.153·29-s − 1.73·31-s + 0.176·32-s + 0.828·34-s + 0.816·35-s − 1.12·37-s + 0.158·40-s + 0.883·41-s − 1.29·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.839006453\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.839006453\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 - 4.82T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 29 | \( 1 + 0.828T + 29T^{2} \) |
| 31 | \( 1 + 9.65T + 31T^{2} \) |
| 37 | \( 1 + 6.82T + 37T^{2} \) |
| 41 | \( 1 - 5.65T + 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 9.31T + 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 + 3.17T + 61T^{2} \) |
| 67 | \( 1 - 1.17T + 67T^{2} \) |
| 71 | \( 1 + 2.82T + 71T^{2} \) |
| 73 | \( 1 - 9.31T + 73T^{2} \) |
| 79 | \( 1 - 2.82T + 79T^{2} \) |
| 83 | \( 1 + 6.82T + 83T^{2} \) |
| 89 | \( 1 + 5.65T + 89T^{2} \) |
| 97 | \( 1 - 8.82T + 97T^{2} \) |
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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
−9.099630985199850900613545259101, −8.184479842425552140327036894934, −7.52015209313273451244555518370, −6.79837747273482454366499359030, −5.56018838888448730508112358227, −5.22053570835672901738345440110, −4.37153796575677075830906830575, −3.38569082367677633454093907295, −2.10533781722412206466063352733, −1.39043206902009212889585974407,
1.39043206902009212889585974407, 2.10533781722412206466063352733, 3.38569082367677633454093907295, 4.37153796575677075830906830575, 5.22053570835672901738345440110, 5.56018838888448730508112358227, 6.79837747273482454366499359030, 7.52015209313273451244555518370, 8.184479842425552140327036894934, 9.099630985199850900613545259101