| L(s) = 1 | − 2.17·2-s + 2.73·4-s − 5-s − 3.90·7-s − 1.59·8-s + 2.17·10-s − 1.59·11-s + 8.49·14-s − 1.99·16-s + 3.76·17-s − 7.91·19-s − 2.73·20-s + 3.46·22-s − 6.22·23-s + 25-s − 10.6·28-s + 5.03·29-s + 0.184·31-s + 7.53·32-s − 8.19·34-s + 3.90·35-s − 1.64·37-s + 17.2·38-s + 1.59·40-s + 10.3·41-s + 6.74·43-s − 4.35·44-s + ⋯ |
| L(s) = 1 | − 1.53·2-s + 1.36·4-s − 0.447·5-s − 1.47·7-s − 0.563·8-s + 0.687·10-s − 0.480·11-s + 2.27·14-s − 0.499·16-s + 0.913·17-s − 1.81·19-s − 0.610·20-s + 0.738·22-s − 1.29·23-s + 0.200·25-s − 2.01·28-s + 0.935·29-s + 0.0332·31-s + 1.33·32-s − 1.40·34-s + 0.660·35-s − 0.271·37-s + 2.79·38-s + 0.251·40-s + 1.61·41-s + 1.02·43-s − 0.655·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.17T + 2T^{2} \) |
| 7 | \( 1 + 3.90T + 7T^{2} \) |
| 11 | \( 1 + 1.59T + 11T^{2} \) |
| 17 | \( 1 - 3.76T + 17T^{2} \) |
| 19 | \( 1 + 7.91T + 19T^{2} \) |
| 23 | \( 1 + 6.22T + 23T^{2} \) |
| 29 | \( 1 - 5.03T + 29T^{2} \) |
| 31 | \( 1 - 0.184T + 31T^{2} \) |
| 37 | \( 1 + 1.64T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 6.74T + 43T^{2} \) |
| 47 | \( 1 - 6.58T + 47T^{2} \) |
| 53 | \( 1 - 5.51T + 53T^{2} \) |
| 59 | \( 1 + 4.32T + 59T^{2} \) |
| 61 | \( 1 + 9.73T + 61T^{2} \) |
| 67 | \( 1 - 6.16T + 67T^{2} \) |
| 71 | \( 1 + 1.24T + 71T^{2} \) |
| 73 | \( 1 - 2.25T + 73T^{2} \) |
| 79 | \( 1 + 4.33T + 79T^{2} \) |
| 83 | \( 1 - 9.12T + 83T^{2} \) |
| 89 | \( 1 + 9.14T + 89T^{2} \) |
| 97 | \( 1 + 9.58T + 97T^{2} \) |
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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
−7.65280750878149996926448870323, −7.10952553700789227943149081817, −6.26205086991406303418659804332, −5.90764912928292678872710528414, −4.52119066048708405399259247721, −3.83660871330002462382940524088, −2.81930099304209459765253704309, −2.15845295610783386502321583769, −0.816245389920147711546190153018, 0,
0.816245389920147711546190153018, 2.15845295610783386502321583769, 2.81930099304209459765253704309, 3.83660871330002462382940524088, 4.52119066048708405399259247721, 5.90764912928292678872710528414, 6.26205086991406303418659804332, 7.10952553700789227943149081817, 7.65280750878149996926448870323
