| L(s) = 1 | + 2-s + 4-s + 1.08·5-s − 0.863·7-s + 8-s + 1.08·10-s − 0.657·11-s + 6.20·13-s − 0.863·14-s + 16-s − 5.69·17-s − 6.45·19-s + 1.08·20-s − 0.657·22-s − 3.81·25-s + 6.20·26-s − 0.863·28-s − 0.101·29-s − 1.54·31-s + 32-s − 5.69·34-s − 0.939·35-s − 7.33·37-s − 6.45·38-s + 1.08·40-s − 6.09·41-s − 5.37·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.486·5-s − 0.326·7-s + 0.353·8-s + 0.344·10-s − 0.198·11-s + 1.71·13-s − 0.230·14-s + 0.250·16-s − 1.38·17-s − 1.48·19-s + 0.243·20-s − 0.140·22-s − 0.763·25-s + 1.21·26-s − 0.163·28-s − 0.0189·29-s − 0.277·31-s + 0.176·32-s − 0.976·34-s − 0.158·35-s − 1.20·37-s − 1.04·38-s + 0.172·40-s − 0.951·41-s − 0.819·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 1.08T + 5T^{2} \) |
| 7 | \( 1 + 0.863T + 7T^{2} \) |
| 11 | \( 1 + 0.657T + 11T^{2} \) |
| 13 | \( 1 - 6.20T + 13T^{2} \) |
| 17 | \( 1 + 5.69T + 17T^{2} \) |
| 19 | \( 1 + 6.45T + 19T^{2} \) |
| 29 | \( 1 + 0.101T + 29T^{2} \) |
| 31 | \( 1 + 1.54T + 31T^{2} \) |
| 37 | \( 1 + 7.33T + 37T^{2} \) |
| 41 | \( 1 + 6.09T + 41T^{2} \) |
| 43 | \( 1 + 5.37T + 43T^{2} \) |
| 47 | \( 1 - 7.84T + 47T^{2} \) |
| 53 | \( 1 - 4.75T + 53T^{2} \) |
| 59 | \( 1 + 5.68T + 59T^{2} \) |
| 61 | \( 1 - 0.224T + 61T^{2} \) |
| 67 | \( 1 - 2.37T + 67T^{2} \) |
| 71 | \( 1 + 6.42T + 71T^{2} \) |
| 73 | \( 1 + 4.17T + 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 + 5.06T + 89T^{2} \) |
| 97 | \( 1 + 9.83T + 97T^{2} \) |
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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
−7.02557004621485548998204121310, −6.51663900029591451174063503944, −6.03627139168815112497550342782, −5.38749642543555911379578068476, −4.44641468189881228994555707575, −3.90529997015894873643353208339, −3.15580388352416853447093877316, −2.15767080393597244492975411474, −1.57428917898071121366584234725, 0,
1.57428917898071121366584234725, 2.15767080393597244492975411474, 3.15580388352416853447093877316, 3.90529997015894873643353208339, 4.44641468189881228994555707575, 5.38749642543555911379578068476, 6.03627139168815112497550342782, 6.51663900029591451174063503944, 7.02557004621485548998204121310
