L(s) = 1 | − 0.586·2-s − 1.65·4-s + 0.842·5-s + 1.23·7-s + 2.14·8-s − 0.494·10-s − 1.47·11-s − 3.26·13-s − 0.725·14-s + 2.05·16-s + 4.93·17-s + 0.260·19-s − 1.39·20-s + 0.861·22-s − 23-s − 4.28·25-s + 1.91·26-s − 2.05·28-s + 29-s − 0.101·31-s − 5.49·32-s − 2.89·34-s + 1.04·35-s − 3.33·37-s − 0.152·38-s + 1.80·40-s + 0.732·41-s + ⋯ |
L(s) = 1 | − 0.414·2-s − 0.828·4-s + 0.376·5-s + 0.468·7-s + 0.757·8-s − 0.156·10-s − 0.443·11-s − 0.904·13-s − 0.194·14-s + 0.514·16-s + 1.19·17-s + 0.0598·19-s − 0.312·20-s + 0.183·22-s − 0.208·23-s − 0.857·25-s + 0.374·26-s − 0.387·28-s + 0.185·29-s − 0.0182·31-s − 0.970·32-s − 0.496·34-s + 0.176·35-s − 0.548·37-s − 0.0248·38-s + 0.285·40-s + 0.114·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 + 0.586T + 2T^{2} \) |
| 5 | \( 1 - 0.842T + 5T^{2} \) |
| 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 + 1.47T + 11T^{2} \) |
| 13 | \( 1 + 3.26T + 13T^{2} \) |
| 17 | \( 1 - 4.93T + 17T^{2} \) |
| 19 | \( 1 - 0.260T + 19T^{2} \) |
| 31 | \( 1 + 0.101T + 31T^{2} \) |
| 37 | \( 1 + 3.33T + 37T^{2} \) |
| 41 | \( 1 - 0.732T + 41T^{2} \) |
| 43 | \( 1 + 5.01T + 43T^{2} \) |
| 47 | \( 1 - 8.71T + 47T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 + 8.76T + 59T^{2} \) |
| 61 | \( 1 + 1.88T + 61T^{2} \) |
| 67 | \( 1 + 3.21T + 67T^{2} \) |
| 71 | \( 1 - 1.72T + 71T^{2} \) |
| 73 | \( 1 + 2.64T + 73T^{2} \) |
| 79 | \( 1 - 0.465T + 79T^{2} \) |
| 83 | \( 1 - 8.34T + 83T^{2} \) |
| 89 | \( 1 - 5.13T + 89T^{2} \) |
| 97 | \( 1 + 6.87T + 97T^{2} \) |
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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
−7.73375833751198043393590579003, −7.39520098198492210452282309649, −6.22109087730296851571216590440, −5.38966346747140602776143224801, −4.98245035264754679482117611234, −4.12939245089958180620831937189, −3.24018649433003913302487903216, −2.16126485165231768044948467772, −1.22537679002315713968916008926, 0,
1.22537679002315713968916008926, 2.16126485165231768044948467772, 3.24018649433003913302487903216, 4.12939245089958180620831937189, 4.98245035264754679482117611234, 5.38966346747140602776143224801, 6.22109087730296851571216590440, 7.39520098198492210452282309649, 7.73375833751198043393590579003