| L(s) = 1 | − 2-s + 2.28·3-s + 4-s + 5-s − 2.28·6-s + 0.279·7-s − 8-s + 2.21·9-s − 10-s − 2.35·11-s + 2.28·12-s + 13-s − 0.279·14-s + 2.28·15-s + 16-s − 6.19·17-s − 2.21·18-s + 2.31·19-s + 20-s + 0.637·21-s + 2.35·22-s + 9.10·23-s − 2.28·24-s + 25-s − 26-s − 1.80·27-s + 0.279·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.31·3-s + 0.5·4-s + 0.447·5-s − 0.931·6-s + 0.105·7-s − 0.353·8-s + 0.736·9-s − 0.316·10-s − 0.710·11-s + 0.658·12-s + 0.277·13-s − 0.0745·14-s + 0.589·15-s + 0.250·16-s − 1.50·17-s − 0.521·18-s + 0.531·19-s + 0.223·20-s + 0.139·21-s + 0.502·22-s + 1.89·23-s − 0.465·24-s + 0.200·25-s − 0.196·26-s − 0.346·27-s + 0.0527·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.417601528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.417601528\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 3 | \( 1 - 2.28T + 3T^{2} \) |
| 7 | \( 1 - 0.279T + 7T^{2} \) |
| 11 | \( 1 + 2.35T + 11T^{2} \) |
| 17 | \( 1 + 6.19T + 17T^{2} \) |
| 19 | \( 1 - 2.31T + 19T^{2} \) |
| 23 | \( 1 - 9.10T + 23T^{2} \) |
| 29 | \( 1 - 7.63T + 29T^{2} \) |
| 37 | \( 1 + 3.29T + 37T^{2} \) |
| 41 | \( 1 + 0.915T + 41T^{2} \) |
| 43 | \( 1 - 5.71T + 43T^{2} \) |
| 47 | \( 1 - 4.89T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 2.55T + 59T^{2} \) |
| 61 | \( 1 - 9.20T + 61T^{2} \) |
| 67 | \( 1 - 6.13T + 67T^{2} \) |
| 71 | \( 1 - 5.72T + 71T^{2} \) |
| 73 | \( 1 + 0.636T + 73T^{2} \) |
| 79 | \( 1 + 0.347T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 - 4.03T + 89T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−8.608536203953215642561123217252, −7.947589774103995184219684092933, −7.11366280468831819285969679228, −6.59245022517307467382592917930, −5.46801413724115821384773570657, −4.64575922891451322485976667936, −3.50321096009900276614073785485, −2.67055606346722121435221115097, −2.19440372324004964350183869174, −0.939370822041732339772581560642,
0.939370822041732339772581560642, 2.19440372324004964350183869174, 2.67055606346722121435221115097, 3.50321096009900276614073785485, 4.64575922891451322485976667936, 5.46801413724115821384773570657, 6.59245022517307467382592917930, 7.11366280468831819285969679228, 7.947589774103995184219684092933, 8.608536203953215642561123217252
