| L(s) = 1 | − 2·2-s + 7·3-s + 4·4-s − 14·6-s + 18·7-s − 8·8-s + 22·9-s + 27·11-s + 28·12-s + 57·13-s − 36·14-s + 16·16-s + 44·17-s − 44·18-s + 152·19-s + 126·21-s − 54·22-s + 152·23-s − 56·24-s − 114·26-s − 35·27-s + 72·28-s − 29·29-s − 173·31-s − 32·32-s + 189·33-s − 88·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.34·3-s + 1/2·4-s − 0.952·6-s + 0.971·7-s − 0.353·8-s + 0.814·9-s + 0.740·11-s + 0.673·12-s + 1.21·13-s − 0.687·14-s + 1/4·16-s + 0.627·17-s − 0.576·18-s + 1.83·19-s + 1.30·21-s − 0.523·22-s + 1.37·23-s − 0.476·24-s − 0.859·26-s − 0.249·27-s + 0.485·28-s − 0.185·29-s − 1.00·31-s − 0.176·32-s + 0.996·33-s − 0.443·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.694278989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.694278989\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 29 | \( 1 + p T \) |
| good | 3 | \( 1 - 7 T + p^{3} T^{2} \) |
| 7 | \( 1 - 18 T + p^{3} T^{2} \) |
| 11 | \( 1 - 27 T + p^{3} T^{2} \) |
| 13 | \( 1 - 57 T + p^{3} T^{2} \) |
| 17 | \( 1 - 44 T + p^{3} T^{2} \) |
| 19 | \( 1 - 8 p T + p^{3} T^{2} \) |
| 23 | \( 1 - 152 T + p^{3} T^{2} \) |
| 31 | \( 1 + 173 T + p^{3} T^{2} \) |
| 37 | \( 1 - 120 T + p^{3} T^{2} \) |
| 41 | \( 1 + 314 T + p^{3} T^{2} \) |
| 43 | \( 1 + 339 T + p^{3} T^{2} \) |
| 47 | \( 1 - 357 T + p^{3} T^{2} \) |
| 53 | \( 1 - 59 T + p^{3} T^{2} \) |
| 59 | \( 1 + 572 T + p^{3} T^{2} \) |
| 61 | \( 1 + 420 T + p^{3} T^{2} \) |
| 67 | \( 1 + 660 T + p^{3} T^{2} \) |
| 71 | \( 1 - 726 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1004 T + p^{3} T^{2} \) |
| 79 | \( 1 - 361 T + p^{3} T^{2} \) |
| 83 | \( 1 - 168 T + p^{3} T^{2} \) |
| 89 | \( 1 - 58 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1206 T + p^{3} T^{2} \) |
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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
−9.023033361697044172518213687628, −8.505340476975161898783946594135, −7.68886156672470684860865825001, −7.21133798568477144104385570898, −5.97326210595678134413803570878, −4.94068577611298834219452285818, −3.59110659985683191283649675123, −3.09601615468710780561849551410, −1.70705876185381951608476608692, −1.13427177783275707165701388237,
1.13427177783275707165701388237, 1.70705876185381951608476608692, 3.09601615468710780561849551410, 3.59110659985683191283649675123, 4.94068577611298834219452285818, 5.97326210595678134413803570878, 7.21133798568477144104385570898, 7.68886156672470684860865825001, 8.505340476975161898783946594135, 9.023033361697044172518213687628
