L(s) = 1 | − 0.223·2-s + 3-s − 1.95·4-s + 1.64·5-s − 0.223·6-s − 7-s + 0.882·8-s + 9-s − 0.367·10-s + 4.86·11-s − 1.95·12-s + 4.77·13-s + 0.223·14-s + 1.64·15-s + 3.70·16-s + 2.25·17-s − 0.223·18-s − 1.51·19-s − 3.20·20-s − 21-s − 1.08·22-s − 1.56·23-s + 0.882·24-s − 2.29·25-s − 1.06·26-s + 27-s + 1.95·28-s + ⋯ |
L(s) = 1 | − 0.157·2-s + 0.577·3-s − 0.975·4-s + 0.735·5-s − 0.0911·6-s − 0.377·7-s + 0.311·8-s + 0.333·9-s − 0.116·10-s + 1.46·11-s − 0.562·12-s + 1.32·13-s + 0.0596·14-s + 0.424·15-s + 0.925·16-s + 0.546·17-s − 0.0526·18-s − 0.348·19-s − 0.717·20-s − 0.218·21-s − 0.231·22-s − 0.326·23-s + 0.180·24-s − 0.458·25-s − 0.209·26-s + 0.192·27-s + 0.368·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.354661252\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.354661252\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 + T \) |
good | 2 | \( 1 + 0.223T + 2T^{2} \) |
| 5 | \( 1 - 1.64T + 5T^{2} \) |
| 11 | \( 1 - 4.86T + 11T^{2} \) |
| 13 | \( 1 - 4.77T + 13T^{2} \) |
| 17 | \( 1 - 2.25T + 17T^{2} \) |
| 19 | \( 1 + 1.51T + 19T^{2} \) |
| 23 | \( 1 + 1.56T + 23T^{2} \) |
| 29 | \( 1 - 5.30T + 29T^{2} \) |
| 31 | \( 1 + 5.90T + 31T^{2} \) |
| 37 | \( 1 - 9.20T + 37T^{2} \) |
| 41 | \( 1 + 6.69T + 41T^{2} \) |
| 43 | \( 1 - 9.36T + 43T^{2} \) |
| 47 | \( 1 + 2.53T + 47T^{2} \) |
| 53 | \( 1 - 6.61T + 53T^{2} \) |
| 59 | \( 1 + 6.16T + 59T^{2} \) |
| 61 | \( 1 - 7.37T + 61T^{2} \) |
| 67 | \( 1 + 0.551T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 - 3.39T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 + 8.33T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 - 5.63T + 97T^{2} \) |
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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
−8.714838991534681179881771121425, −7.940684038978567020267581769340, −6.98486856164504786296103009608, −6.08967710290182559440176565141, −5.69878342358698736982025675785, −4.38567925533801212123055943299, −3.91055455486864352851905214548, −3.11398702019906515215469847350, −1.77405194916622801595493696954, −0.965671192613006786368568539119,
0.965671192613006786368568539119, 1.77405194916622801595493696954, 3.11398702019906515215469847350, 3.91055455486864352851905214548, 4.38567925533801212123055943299, 5.69878342358698736982025675785, 6.08967710290182559440176565141, 6.98486856164504786296103009608, 7.940684038978567020267581769340, 8.714838991534681179881771121425