| L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s − 11-s + 4.68·13-s + 14-s + 16-s + 0.292·17-s + 6.51·19-s − 20-s + 22-s − 2.85·23-s + 25-s − 4.68·26-s − 28-s + 1.43·29-s + 0.978·31-s − 32-s − 0.292·34-s + 35-s + 0.853·37-s − 6.51·38-s + 40-s + 6.22·41-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 + 1.29·13-s + 0.267·14-s + 0.250·16-s + 0.0709·17-s + 1.49·19-s − 0.223·20-s + 0.213·22-s − 0.595·23-s + 0.200·25-s − 0.918·26-s − 0.188·28-s + 0.267·29-s + 0.175·31-s − 0.176·32-s − 0.0502·34-s + 0.169·35-s + 0.140·37-s − 1.05·38-s + 0.158·40-s + 0.972·41-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
| \(L(1)\) |
\(\approx\) |
\(1.242337311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.242337311\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4.68T + 13T^{2} \) |
| 17 | \( 1 - 0.292T + 17T^{2} \) |
| 19 | \( 1 - 6.51T + 19T^{2} \) |
| 23 | \( 1 + 2.85T + 23T^{2} \) |
| 29 | \( 1 - 1.43T + 29T^{2} \) |
| 31 | \( 1 - 0.978T + 31T^{2} \) |
| 37 | \( 1 - 0.853T + 37T^{2} \) |
| 41 | \( 1 - 6.22T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 9.95T + 47T^{2} \) |
| 53 | \( 1 + 5.43T + 53T^{2} \) |
| 59 | \( 1 - 9.37T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 + 0.585T + 67T^{2} \) |
| 71 | \( 1 - 0.335T + 71T^{2} \) |
| 73 | \( 1 + 3.70T + 73T^{2} \) |
| 79 | \( 1 + 2.51T + 79T^{2} \) |
| 83 | \( 1 - 1.70T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 - 9.10T + 97T^{2} \) |
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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
−7.900541596082541343454508279587, −7.48060808125544692999137989541, −6.62899616370167223782100440452, −5.98922604176304663676044977178, −5.28383837323806687741950422588, −4.21641061333996531826932461763, −3.43264114714875048809737981738, −2.78706925799825349980546117231, −1.56751638773802829439853075035, −0.66264159211278378958030835745,
0.66264159211278378958030835745, 1.56751638773802829439853075035, 2.78706925799825349980546117231, 3.43264114714875048809737981738, 4.21641061333996531826932461763, 5.28383837323806687741950422588, 5.98922604176304663676044977178, 6.62899616370167223782100440452, 7.48060808125544692999137989541, 7.900541596082541343454508279587
