| L(s) = 1 | − 3.25·3-s − 0.557·5-s + 7-s + 7.62·9-s + 5.17·11-s − 2·13-s + 1.81·15-s + 17-s − 3.25·21-s + 6.51·23-s − 4.68·25-s − 15.0·27-s − 3.54·29-s − 4.84·31-s − 16.8·33-s − 0.557·35-s + 3.17·37-s + 6.51·39-s − 5.32·41-s + 4.59·43-s − 4.24·45-s − 0.746·47-s + 49-s − 3.25·51-s − 11.5·53-s − 2.88·55-s − 10.3·59-s + ⋯ |
| L(s) = 1 | − 1.88·3-s − 0.249·5-s + 0.377·7-s + 2.54·9-s + 1.56·11-s − 0.554·13-s + 0.468·15-s + 0.242·17-s − 0.711·21-s + 1.35·23-s − 0.937·25-s − 2.89·27-s − 0.658·29-s − 0.870·31-s − 2.93·33-s − 0.0942·35-s + 0.522·37-s + 1.04·39-s − 0.831·41-s + 0.701·43-s − 0.633·45-s − 0.108·47-s + 0.142·49-s − 0.456·51-s − 1.58·53-s − 0.389·55-s − 1.34·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 3.25T + 3T^{2} \) |
| 5 | \( 1 + 0.557T + 5T^{2} \) |
| 11 | \( 1 - 5.17T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 6.51T + 23T^{2} \) |
| 29 | \( 1 + 3.54T + 29T^{2} \) |
| 31 | \( 1 + 4.84T + 31T^{2} \) |
| 37 | \( 1 - 3.17T + 37T^{2} \) |
| 41 | \( 1 + 5.32T + 41T^{2} \) |
| 43 | \( 1 - 4.59T + 43T^{2} \) |
| 47 | \( 1 + 0.746T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 1.25T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 + 16.1T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 - 0.746T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 - 2.20T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−7.24263071901513358567161576661, −6.80343912911852215654854882192, −6.05676644215075193791684550063, −5.51878283625819937903702156860, −4.73367660104303160426679118598, −4.24278815922949941127460464922, −3.38700443305672251089112107293, −1.79565610759065507372465169700, −1.12003992879299062320825316111, 0,
1.12003992879299062320825316111, 1.79565610759065507372465169700, 3.38700443305672251089112107293, 4.24278815922949941127460464922, 4.73367660104303160426679118598, 5.51878283625819937903702156860, 6.05676644215075193791684550063, 6.80343912911852215654854882192, 7.24263071901513358567161576661
