| L(s) = 1 | − 3.12·3-s − 2.64·5-s − 7-s + 6.76·9-s − 2·11-s − 2.48·13-s + 8.24·15-s + 5.60·17-s − 2.96·19-s + 3.12·21-s − 23-s + 1.96·25-s − 11.7·27-s + 1.51·29-s + 1.12·31-s + 6.24·33-s + 2.64·35-s + 8.24·37-s + 7.76·39-s + 1.76·41-s + 12.4·43-s − 17.8·45-s + 6.40·47-s + 49-s − 17.5·51-s + 6.31·53-s + 5.28·55-s + ⋯ |
| L(s) = 1 | − 1.80·3-s − 1.18·5-s − 0.377·7-s + 2.25·9-s − 0.603·11-s − 0.689·13-s + 2.13·15-s + 1.36·17-s − 0.681·19-s + 0.681·21-s − 0.208·23-s + 0.393·25-s − 2.26·27-s + 0.281·29-s + 0.202·31-s + 1.08·33-s + 0.446·35-s + 1.35·37-s + 1.24·39-s + 0.275·41-s + 1.90·43-s − 2.66·45-s + 0.934·47-s + 0.142·49-s − 2.45·51-s + 0.866·53-s + 0.711·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 3.12T + 3T^{2} \) |
| 5 | \( 1 + 2.64T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 2.48T + 13T^{2} \) |
| 17 | \( 1 - 5.60T + 17T^{2} \) |
| 19 | \( 1 + 2.96T + 19T^{2} \) |
| 29 | \( 1 - 1.51T + 29T^{2} \) |
| 31 | \( 1 - 1.12T + 31T^{2} \) |
| 37 | \( 1 - 8.24T + 37T^{2} \) |
| 41 | \( 1 - 1.76T + 41T^{2} \) |
| 43 | \( 1 - 12.4T + 43T^{2} \) |
| 47 | \( 1 - 6.40T + 47T^{2} \) |
| 53 | \( 1 - 6.31T + 53T^{2} \) |
| 59 | \( 1 + 3.67T + 59T^{2} \) |
| 61 | \( 1 - 7.60T + 61T^{2} \) |
| 67 | \( 1 + 9.03T + 67T^{2} \) |
| 71 | \( 1 - 6.01T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 + 6.31T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−8.262887551139839055086868120540, −7.47843163665119043634361414054, −7.07512127342657986718482072498, −5.94731934040156138969884239944, −5.57578345644431362255710603754, −4.49945688723513186882385352243, −4.05037121790960579365928756793, −2.71348085182481539850404495881, −0.965448938968600463426713637226, 0,
0.965448938968600463426713637226, 2.71348085182481539850404495881, 4.05037121790960579365928756793, 4.49945688723513186882385352243, 5.57578345644431362255710603754, 5.94731934040156138969884239944, 7.07512127342657986718482072498, 7.47843163665119043634361414054, 8.262887551139839055086868120540
