| L(s) = 1 | − 2.43·2-s − 0.763·3-s + 3.93·4-s + 1.56·5-s + 1.86·6-s + 2.19·7-s − 4.70·8-s − 2.41·9-s − 3.81·10-s + 11-s − 3.00·12-s + 2.36·13-s − 5.33·14-s − 1.19·15-s + 3.58·16-s − 7.19·17-s + 5.88·18-s − 5.00·19-s + 6.16·20-s − 1.67·21-s − 2.43·22-s + 5.48·23-s + 3.59·24-s − 2.54·25-s − 5.75·26-s + 4.13·27-s + 8.61·28-s + ⋯ |
| L(s) = 1 | − 1.72·2-s − 0.441·3-s + 1.96·4-s + 0.701·5-s + 0.759·6-s + 0.828·7-s − 1.66·8-s − 0.805·9-s − 1.20·10-s + 0.301·11-s − 0.866·12-s + 0.654·13-s − 1.42·14-s − 0.309·15-s + 0.896·16-s − 1.74·17-s + 1.38·18-s − 1.14·19-s + 1.37·20-s − 0.365·21-s − 0.519·22-s + 1.14·23-s + 0.732·24-s − 0.508·25-s − 1.12·26-s + 0.796·27-s + 1.62·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 547 | \( 1 - T \) |
| good | 2 | \( 1 + 2.43T + 2T^{2} \) |
| 3 | \( 1 + 0.763T + 3T^{2} \) |
| 5 | \( 1 - 1.56T + 5T^{2} \) |
| 7 | \( 1 - 2.19T + 7T^{2} \) |
| 13 | \( 1 - 2.36T + 13T^{2} \) |
| 17 | \( 1 + 7.19T + 17T^{2} \) |
| 19 | \( 1 + 5.00T + 19T^{2} \) |
| 23 | \( 1 - 5.48T + 23T^{2} \) |
| 29 | \( 1 - 3.08T + 29T^{2} \) |
| 31 | \( 1 - 0.979T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + 6.32T + 41T^{2} \) |
| 43 | \( 1 - 12.6T + 43T^{2} \) |
| 47 | \( 1 - 8.78T + 47T^{2} \) |
| 53 | \( 1 + 0.712T + 53T^{2} \) |
| 59 | \( 1 - 5.94T + 59T^{2} \) |
| 61 | \( 1 + 8.95T + 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 5.23T + 73T^{2} \) |
| 79 | \( 1 + 2.02T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 + 9.97T + 89T^{2} \) |
| 97 | \( 1 + 9.49T + 97T^{2} \) |
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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
−8.084245636250205507371956872436, −6.90821199410627101880958866242, −6.64702242109389423951590303351, −5.85144252942115284773309205866, −5.03903563108982094433293228664, −4.07083168185117694567064312593, −2.62631454784641653045779269129, −2.03307595671006252680434230617, −1.15243126944000429498728074534, 0,
1.15243126944000429498728074534, 2.03307595671006252680434230617, 2.62631454784641653045779269129, 4.07083168185117694567064312593, 5.03903563108982094433293228664, 5.85144252942115284773309205866, 6.64702242109389423951590303351, 6.90821199410627101880958866242, 8.084245636250205507371956872436
