| L(s) = 1 | + 0.618·2-s − 3.23·3-s − 1.61·4-s − 2.00·6-s − 2.23·8-s + 7.47·9-s − 0.236·11-s + 5.23·12-s + 1.23·13-s + 1.85·16-s + 2.47·17-s + 4.61·18-s + 4.47·19-s − 0.145·22-s − 6.23·23-s + 7.23·24-s + 0.763·26-s − 14.4·27-s + 5·29-s − 3.70·31-s + 5.61·32-s + 0.763·33-s + 1.52·34-s − 12.0·36-s − 3·37-s + 2.76·38-s − 4.00·39-s + ⋯ |
| L(s) = 1 | + 0.437·2-s − 1.86·3-s − 0.809·4-s − 0.816·6-s − 0.790·8-s + 2.49·9-s − 0.0711·11-s + 1.51·12-s + 0.342·13-s + 0.463·16-s + 0.599·17-s + 1.08·18-s + 1.02·19-s − 0.0311·22-s − 1.30·23-s + 1.47·24-s + 0.149·26-s − 2.78·27-s + 0.928·29-s − 0.666·31-s + 0.993·32-s + 0.132·33-s + 0.262·34-s − 2.01·36-s − 0.493·37-s + 0.448·38-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 3 | \( 1 + 3.23T + 3T^{2} \) |
| 11 | \( 1 + 0.236T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 23 | \( 1 + 6.23T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + 3.70T + 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 + 4.76T + 41T^{2} \) |
| 43 | \( 1 + 1.76T + 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 9.70T + 61T^{2} \) |
| 67 | \( 1 - 4.23T + 67T^{2} \) |
| 71 | \( 1 - 8.70T + 71T^{2} \) |
| 73 | \( 1 + 8.76T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + 7.70T + 83T^{2} \) |
| 89 | \( 1 + 17.2T + 89T^{2} \) |
| 97 | \( 1 - 5.23T + 97T^{2} \) |
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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
−9.761063150317308201526623951396, −8.476369265678107975394008411230, −7.47547604306337529414069111501, −6.46313891296476344243431476870, −5.74806949845994802134984751338, −5.17321536099051064306185975283, −4.39517435935058781015767495428, −3.44006352602964746714616065249, −1.31710662654464286407497925571, 0,
1.31710662654464286407497925571, 3.44006352602964746714616065249, 4.39517435935058781015767495428, 5.17321536099051064306185975283, 5.74806949845994802134984751338, 6.46313891296476344243431476870, 7.47547604306337529414069111501, 8.476369265678107975394008411230, 9.761063150317308201526623951396
