| L(s) = 1 | − 9·3-s − 5·5-s − 7·7-s + 54·9-s − 55·11-s + 69·13-s + 45·15-s + 113·17-s + 126·19-s + 63·21-s − 102·23-s + 25·25-s − 243·27-s + 81·29-s + 176·31-s + 495·33-s + 35·35-s − 254·37-s − 621·39-s − 184·41-s + 230·43-s − 270·45-s − 187·47-s + 49·49-s − 1.01e3·51-s + 488·53-s + 275·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.447·5-s − 0.377·7-s + 2·9-s − 1.50·11-s + 1.47·13-s + 0.774·15-s + 1.61·17-s + 1.52·19-s + 0.654·21-s − 0.924·23-s + 1/5·25-s − 1.73·27-s + 0.518·29-s + 1.01·31-s + 2.61·33-s + 0.169·35-s − 1.12·37-s − 2.54·39-s − 0.700·41-s + 0.815·43-s − 0.894·45-s − 0.580·47-s + 1/7·49-s − 2.79·51-s + 1.26·53-s + 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9126995100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9126995100\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 + p T \) |
| good | 3 | \( 1 + p^{2} T + p^{3} T^{2} \) |
| 11 | \( 1 + 5 p T + p^{3} T^{2} \) |
| 13 | \( 1 - 69 T + p^{3} T^{2} \) |
| 17 | \( 1 - 113 T + p^{3} T^{2} \) |
| 19 | \( 1 - 126 T + p^{3} T^{2} \) |
| 23 | \( 1 + 102 T + p^{3} T^{2} \) |
| 29 | \( 1 - 81 T + p^{3} T^{2} \) |
| 31 | \( 1 - 176 T + p^{3} T^{2} \) |
| 37 | \( 1 + 254 T + p^{3} T^{2} \) |
| 41 | \( 1 + 184 T + p^{3} T^{2} \) |
| 43 | \( 1 - 230 T + p^{3} T^{2} \) |
| 47 | \( 1 + 187 T + p^{3} T^{2} \) |
| 53 | \( 1 - 488 T + p^{3} T^{2} \) |
| 59 | \( 1 + 388 T + p^{3} T^{2} \) |
| 61 | \( 1 - 728 T + p^{3} T^{2} \) |
| 67 | \( 1 - 96 T + p^{3} T^{2} \) |
| 71 | \( 1 - 8 T + p^{3} T^{2} \) |
| 73 | \( 1 + 994 T + p^{3} T^{2} \) |
| 79 | \( 1 - 337 T + p^{3} T^{2} \) |
| 83 | \( 1 + 188 T + p^{3} T^{2} \) |
| 89 | \( 1 + 884 T + p^{3} T^{2} \) |
| 97 | \( 1 + 451 T + p^{3} T^{2} \) |
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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
−8.503462384823202082266769698742, −7.76257391738573640273714257029, −7.06168280910691230200118891393, −6.10237939799458671424400636598, −5.55931149777965131903998694220, −5.01748909838832286554642260223, −3.88401781444095828384369403144, −3.02656884383407968200397776376, −1.29882337250828102583146432781, −0.53756056886296316572197172447,
0.53756056886296316572197172447, 1.29882337250828102583146432781, 3.02656884383407968200397776376, 3.88401781444095828384369403144, 5.01748909838832286554642260223, 5.55931149777965131903998694220, 6.10237939799458671424400636598, 7.06168280910691230200118891393, 7.76257391738573640273714257029, 8.503462384823202082266769698742
