| L(s) = 1 | + 2.74·2-s − 0.483·4-s − 19.4·5-s + 7.48·7-s − 23.2·8-s − 53.4·10-s − 22.8·11-s − 13·13-s + 20.5·14-s − 59.8·16-s − 67.0·17-s + 16.5·19-s + 9.41·20-s − 62.7·22-s + 175.·23-s + 254.·25-s − 35.6·26-s − 3.61·28-s − 291.·29-s + 117.·31-s + 21.8·32-s − 183.·34-s − 145.·35-s − 154.·37-s + 45.2·38-s + 453.·40-s + 251.·41-s + ⋯ |
| L(s) = 1 | + 0.969·2-s − 0.0604·4-s − 1.74·5-s + 0.404·7-s − 1.02·8-s − 1.68·10-s − 0.627·11-s − 0.277·13-s + 0.391·14-s − 0.935·16-s − 0.956·17-s + 0.199·19-s + 0.105·20-s − 0.608·22-s + 1.59·23-s + 2.03·25-s − 0.268·26-s − 0.0244·28-s − 1.86·29-s + 0.679·31-s + 0.120·32-s − 0.927·34-s − 0.704·35-s − 0.687·37-s + 0.193·38-s + 1.79·40-s + 0.958·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 - 2.74T + 8T^{2} \) |
| 5 | \( 1 + 19.4T + 125T^{2} \) |
| 7 | \( 1 - 7.48T + 343T^{2} \) |
| 11 | \( 1 + 22.8T + 1.33e3T^{2} \) |
| 17 | \( 1 + 67.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 16.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 175.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 291.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 117.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 154.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 251.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 502.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 281.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 366.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 79.6T + 2.05e5T^{2} \) |
| 61 | \( 1 + 194.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 400.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 528.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 734.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 113.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 933.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.19e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 557.T + 9.12e5T^{2} \) |
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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
−12.63377013402291728405151092866, −11.63886287583979460261011077800, −10.95812536459132635329218249832, −9.108916971155018778334359686492, −8.063073205285482351626722771698, −6.97106369438775344828064583840, −5.17572821030091735397467274800, −4.30386547431927186983213540636, −3.13310505933253413178456107533, 0,
3.13310505933253413178456107533, 4.30386547431927186983213540636, 5.17572821030091735397467274800, 6.97106369438775344828064583840, 8.063073205285482351626722771698, 9.108916971155018778334359686492, 10.95812536459132635329218249832, 11.63886287583979460261011077800, 12.63377013402291728405151092866
