| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s + 11-s − 4.74·13-s − 14-s + 16-s − 4.74·17-s − 20-s + 22-s + 6.74·23-s + 25-s − 4.74·26-s − 28-s − 2·29-s + 32-s − 4.74·34-s + 35-s + 8.74·37-s − 40-s − 6·41-s + 4·43-s + 44-s + 6.74·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s − 1.31·13-s − 0.267·14-s + 0.250·16-s − 1.15·17-s − 0.223·20-s + 0.213·22-s + 1.40·23-s + 0.200·25-s − 0.930·26-s − 0.188·28-s − 0.371·29-s + 0.176·32-s − 0.813·34-s + 0.169·35-s + 1.43·37-s − 0.158·40-s − 0.937·41-s + 0.609·43-s + 0.150·44-s + 0.994·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.386608266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.386608266\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + 4.74T + 13T^{2} \) |
| 17 | \( 1 + 4.74T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 6.74T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 8.74T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 1.25T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 4.74T + 89T^{2} \) |
| 97 | \( 1 + 7.48T + 97T^{2} \) |
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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
−7.72852876427675319310153161820, −7.09597438178479917992950539120, −6.66452054258638000614625511292, −5.79416072819502012093045576829, −4.94217470702336875362142336315, −4.45707599956264882745400231796, −3.64001199544440805396990302866, −2.79283853617632354207624939887, −2.11766708283321327541829782228, −0.68426280617406574793262842056,
0.68426280617406574793262842056, 2.11766708283321327541829782228, 2.79283853617632354207624939887, 3.64001199544440805396990302866, 4.45707599956264882745400231796, 4.94217470702336875362142336315, 5.79416072819502012093045576829, 6.66452054258638000614625511292, 7.09597438178479917992950539120, 7.72852876427675319310153161820
