| L(s) = 1 | − 1.17·2-s − 0.624·4-s − 0.178·5-s − 0.0809·7-s + 3.07·8-s + 0.209·10-s + 4.30·11-s + 1.10·13-s + 0.0948·14-s − 2.35·16-s + 3.06·17-s − 2.15·19-s + 0.111·20-s − 5.04·22-s − 9.15·23-s − 4.96·25-s − 1.29·26-s + 0.0505·28-s + 6.17·29-s − 9.82·31-s − 3.38·32-s − 3.59·34-s + 0.0144·35-s + 4.08·37-s + 2.53·38-s − 0.550·40-s − 4.58·41-s + ⋯ |
| L(s) = 1 | − 0.829·2-s − 0.312·4-s − 0.0799·5-s − 0.0305·7-s + 1.08·8-s + 0.0662·10-s + 1.29·11-s + 0.305·13-s + 0.0253·14-s − 0.589·16-s + 0.743·17-s − 0.495·19-s + 0.0249·20-s − 1.07·22-s − 1.90·23-s − 0.993·25-s − 0.253·26-s + 0.00955·28-s + 1.14·29-s − 1.76·31-s − 0.599·32-s − 0.616·34-s + 0.00244·35-s + 0.671·37-s + 0.410·38-s − 0.0870·40-s − 0.716·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4527 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 2 | \( 1 + 1.17T + 2T^{2} \) |
| 5 | \( 1 + 0.178T + 5T^{2} \) |
| 7 | \( 1 + 0.0809T + 7T^{2} \) |
| 11 | \( 1 - 4.30T + 11T^{2} \) |
| 13 | \( 1 - 1.10T + 13T^{2} \) |
| 17 | \( 1 - 3.06T + 17T^{2} \) |
| 19 | \( 1 + 2.15T + 19T^{2} \) |
| 23 | \( 1 + 9.15T + 23T^{2} \) |
| 29 | \( 1 - 6.17T + 29T^{2} \) |
| 31 | \( 1 + 9.82T + 31T^{2} \) |
| 37 | \( 1 - 4.08T + 37T^{2} \) |
| 41 | \( 1 + 4.58T + 41T^{2} \) |
| 43 | \( 1 - 2.03T + 43T^{2} \) |
| 47 | \( 1 - 5.12T + 47T^{2} \) |
| 53 | \( 1 + 2.69T + 53T^{2} \) |
| 59 | \( 1 + 9.21T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 - 5.62T + 67T^{2} \) |
| 71 | \( 1 - 4.88T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 17.4T + 79T^{2} \) |
| 83 | \( 1 - 2.48T + 83T^{2} \) |
| 89 | \( 1 + 5.57T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−7.999070831793428590461208049349, −7.59147137370021142155447513628, −6.51319929767492872256384824567, −5.98193951524113366624087406897, −4.93618531906520041730773696308, −4.02928185735296617668073574452, −3.60705799250809614027606013648, −2.04707217752764682808334174738, −1.26719522784101129221609363480, 0,
1.26719522784101129221609363480, 2.04707217752764682808334174738, 3.60705799250809614027606013648, 4.02928185735296617668073574452, 4.93618531906520041730773696308, 5.98193951524113366624087406897, 6.51319929767492872256384824567, 7.59147137370021142155447513628, 7.999070831793428590461208049349
