| L(s) = 1 | + 1.39·2-s + 1.69·3-s − 0.0663·4-s − 3.04·5-s + 2.35·6-s − 2.87·8-s − 0.132·9-s − 4.22·10-s + 1.09·11-s − 0.112·12-s − 3.88·13-s − 5.15·15-s − 3.86·16-s − 0.753·17-s − 0.184·18-s − 5.15·19-s + 0.201·20-s + 1.51·22-s + 23-s − 4.86·24-s + 4.25·25-s − 5.39·26-s − 5.30·27-s + 0.177·29-s − 7.16·30-s + 0.549·31-s + 0.374·32-s + ⋯ |
| L(s) = 1 | + 0.983·2-s + 0.977·3-s − 0.0331·4-s − 1.36·5-s + 0.961·6-s − 1.01·8-s − 0.0443·9-s − 1.33·10-s + 0.328·11-s − 0.0324·12-s − 1.07·13-s − 1.32·15-s − 0.965·16-s − 0.182·17-s − 0.0435·18-s − 1.18·19-s + 0.0450·20-s + 0.323·22-s + 0.208·23-s − 0.993·24-s + 0.850·25-s − 1.05·26-s − 1.02·27-s + 0.0330·29-s − 1.30·30-s + 0.0987·31-s + 0.0662·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 1.39T + 2T^{2} \) |
| 3 | \( 1 - 1.69T + 3T^{2} \) |
| 5 | \( 1 + 3.04T + 5T^{2} \) |
| 11 | \( 1 - 1.09T + 11T^{2} \) |
| 13 | \( 1 + 3.88T + 13T^{2} \) |
| 17 | \( 1 + 0.753T + 17T^{2} \) |
| 19 | \( 1 + 5.15T + 19T^{2} \) |
| 29 | \( 1 - 0.177T + 29T^{2} \) |
| 31 | \( 1 - 0.549T + 31T^{2} \) |
| 37 | \( 1 - 4.06T + 37T^{2} \) |
| 41 | \( 1 + 9.86T + 41T^{2} \) |
| 43 | \( 1 - 2.33T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 - 0.897T + 53T^{2} \) |
| 59 | \( 1 + 2.07T + 59T^{2} \) |
| 61 | \( 1 + 9.72T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 + 15.7T + 71T^{2} \) |
| 73 | \( 1 + 6.86T + 73T^{2} \) |
| 79 | \( 1 + 2.21T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 + 8.96T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−9.095581709390719620218229361813, −8.627700673571755860088603502137, −7.80526419870260596700502022604, −7.00325798319531289138229888625, −5.87944550553777530534578704449, −4.68546788389912702414822008088, −4.11221530369834627691913749065, −3.29706083497560003356974032610, −2.43020249937999924373534690031, 0,
2.43020249937999924373534690031, 3.29706083497560003356974032610, 4.11221530369834627691913749065, 4.68546788389912702414822008088, 5.87944550553777530534578704449, 7.00325798319531289138229888625, 7.80526419870260596700502022604, 8.627700673571755860088603502137, 9.095581709390719620218229361813
