| L(s) = 1 | − 2·2-s + 9.71·3-s + 4·4-s − 19.4·6-s + 33.1·7-s − 8·8-s + 67.3·9-s + 1.59·11-s + 38.8·12-s − 15.1·13-s − 66.2·14-s + 16·16-s − 63.9·17-s − 134.·18-s − 110.·19-s + 321.·21-s − 3.19·22-s + 23·23-s − 77.6·24-s + 30.2·26-s + 391.·27-s + 132.·28-s + 191.·29-s + 292.·31-s − 32·32-s + 15.5·33-s + 127.·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.86·3-s + 0.5·4-s − 1.32·6-s + 1.78·7-s − 0.353·8-s + 2.49·9-s + 0.0437·11-s + 0.934·12-s − 0.322·13-s − 1.26·14-s + 0.250·16-s − 0.912·17-s − 1.76·18-s − 1.33·19-s + 3.34·21-s − 0.0309·22-s + 0.208·23-s − 0.660·24-s + 0.228·26-s + 2.79·27-s + 0.894·28-s + 1.22·29-s + 1.69·31-s − 0.176·32-s + 0.0818·33-s + 0.645·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.083873092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.083873092\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 9.71T + 27T^{2} \) |
| 7 | \( 1 - 33.1T + 343T^{2} \) |
| 11 | \( 1 - 1.59T + 1.33e3T^{2} \) |
| 13 | \( 1 + 15.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 63.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 110.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 191.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 292.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 62.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 296.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 90.3T + 7.95e4T^{2} \) |
| 47 | \( 1 - 142.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 589.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 175.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 809.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 431.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 453.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 111.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.17e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 95.0T + 5.71e5T^{2} \) |
| 89 | \( 1 - 836.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 599.T + 9.12e5T^{2} \) |
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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
−8.946647439642682488384504955826, −8.684620582305867073588887775535, −8.010733833621020769359914032360, −7.43126199611406028523774238065, −6.45562290088254723466661051933, −4.71986391621217752105552784405, −4.19865109645254006597786473984, −2.70194371641779890499938308996, −2.12582358353521947077132908313, −1.16824705987822713060943279537,
1.16824705987822713060943279537, 2.12582358353521947077132908313, 2.70194371641779890499938308996, 4.19865109645254006597786473984, 4.71986391621217752105552784405, 6.45562290088254723466661051933, 7.43126199611406028523774238065, 8.010733833621020769359914032360, 8.684620582305867073588887775535, 8.946647439642682488384504955826
