L(s) = 1 | − 0.486·3-s + 3.55·5-s + 0.471·7-s − 2.76·9-s − 0.531·11-s + 4.46·13-s − 1.72·15-s − 4.83·17-s − 7.53·19-s − 0.229·21-s − 6.36·23-s + 7.64·25-s + 2.80·27-s − 0.479·29-s + 2.79·31-s + 0.258·33-s + 1.67·35-s + 10.7·37-s − 2.17·39-s − 0.506·41-s − 10.2·43-s − 9.82·45-s − 0.860·47-s − 6.77·49-s + 2.35·51-s − 6.48·53-s − 1.88·55-s + ⋯ |
L(s) = 1 | − 0.280·3-s + 1.59·5-s + 0.178·7-s − 0.921·9-s − 0.160·11-s + 1.23·13-s − 0.446·15-s − 1.17·17-s − 1.72·19-s − 0.0500·21-s − 1.32·23-s + 1.52·25-s + 0.539·27-s − 0.0890·29-s + 0.501·31-s + 0.0449·33-s + 0.283·35-s + 1.76·37-s − 0.347·39-s − 0.0790·41-s − 1.55·43-s − 1.46·45-s − 0.125·47-s − 0.968·49-s + 0.329·51-s − 0.891·53-s − 0.254·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 751 | \( 1 + T \) |
good | 3 | \( 1 + 0.486T + 3T^{2} \) |
| 5 | \( 1 - 3.55T + 5T^{2} \) |
| 7 | \( 1 - 0.471T + 7T^{2} \) |
| 11 | \( 1 + 0.531T + 11T^{2} \) |
| 13 | \( 1 - 4.46T + 13T^{2} \) |
| 17 | \( 1 + 4.83T + 17T^{2} \) |
| 19 | \( 1 + 7.53T + 19T^{2} \) |
| 23 | \( 1 + 6.36T + 23T^{2} \) |
| 29 | \( 1 + 0.479T + 29T^{2} \) |
| 31 | \( 1 - 2.79T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 + 0.506T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 0.860T + 47T^{2} \) |
| 53 | \( 1 + 6.48T + 53T^{2} \) |
| 59 | \( 1 + 3.90T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 + 9.72T + 67T^{2} \) |
| 71 | \( 1 + 7.91T + 71T^{2} \) |
| 73 | \( 1 - 4.56T + 73T^{2} \) |
| 79 | \( 1 - 2.55T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−7.958610401281914468745661563084, −6.45174937078603446579638320940, −6.32933761744253219124613120645, −5.86417170589453114322390841900, −4.89441383570114807608993806120, −4.21976098533947441253186066398, −3.01885966383472986488512004825, −2.19625144199057162574341632591, −1.55060399552044853317605383267, 0,
1.55060399552044853317605383267, 2.19625144199057162574341632591, 3.01885966383472986488512004825, 4.21976098533947441253186066398, 4.89441383570114807608993806120, 5.86417170589453114322390841900, 6.32933761744253219124613120645, 6.45174937078603446579638320940, 7.958610401281914468745661563084