| L(s) = 1 | − 0.149·2-s − 7.97·4-s − 20.1·7-s + 2.38·8-s + 9.89·11-s + 11.8·13-s + 3.00·14-s + 63.4·16-s − 6.09·17-s − 62.6·19-s − 1.47·22-s + 12.0·23-s − 1.76·26-s + 160.·28-s − 140.·29-s + 178.·31-s − 28.5·32-s + 0.908·34-s + 216.·37-s + 9.34·38-s − 411.·41-s + 48.9·43-s − 78.9·44-s − 1.80·46-s + 615.·47-s + 62.7·49-s − 94.3·52-s + ⋯ |
| L(s) = 1 | − 0.0527·2-s − 0.997·4-s − 1.08·7-s + 0.105·8-s + 0.271·11-s + 0.252·13-s + 0.0573·14-s + 0.991·16-s − 0.0869·17-s − 0.756·19-s − 0.0143·22-s + 0.109·23-s − 0.0133·26-s + 1.08·28-s − 0.900·29-s + 1.03·31-s − 0.157·32-s + 0.00458·34-s + 0.960·37-s + 0.0398·38-s − 1.56·41-s + 0.173·43-s − 0.270·44-s − 0.00577·46-s + 1.90·47-s + 0.182·49-s − 0.251·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.149T + 8T^{2} \) |
| 7 | \( 1 + 20.1T + 343T^{2} \) |
| 11 | \( 1 - 9.89T + 1.33e3T^{2} \) |
| 13 | \( 1 - 11.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 6.09T + 4.91e3T^{2} \) |
| 19 | \( 1 + 62.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 12.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 140.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 178.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 216.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 411.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 48.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 615.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 705.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 494.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 666.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 277.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 239.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 919.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 516.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 652.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 543.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.13e3T + 9.12e5T^{2} \) |
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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
−8.616540701714534723077629675352, −7.71443410901139674332587331243, −6.73512291154903168534018680233, −6.03808660550981243948937428741, −5.17465529905015720410219000057, −4.14632358812579277195262968445, −3.58945866188328774193846839870, −2.46148096740028368364919605484, −0.987893137986791053856275379840, 0,
0.987893137986791053856275379840, 2.46148096740028368364919605484, 3.58945866188328774193846839870, 4.14632358812579277195262968445, 5.17465529905015720410219000057, 6.03808660550981243948937428741, 6.73512291154903168534018680233, 7.71443410901139674332587331243, 8.616540701714534723077629675352
