| L(s) = 1 | − 1.64·5-s − 7-s + 5.29·11-s − 5.64·17-s + 19-s + 5.29·23-s − 2.29·25-s + 4.35·29-s − 6·31-s + 1.64·35-s − 5.29·37-s − 7.29·41-s + 9.29·43-s + 9.64·47-s + 49-s + 8.35·53-s − 8.70·55-s + 8·59-s − 1.29·61-s − 3.29·67-s − 6.93·71-s − 2·73-s − 5.29·77-s + 7.29·79-s − 0.937·83-s + 9.29·85-s − 7.29·89-s + ⋯ |
| L(s) = 1 | − 0.736·5-s − 0.377·7-s + 1.59·11-s − 1.36·17-s + 0.229·19-s + 1.10·23-s − 0.458·25-s + 0.808·29-s − 1.07·31-s + 0.278·35-s − 0.869·37-s − 1.13·41-s + 1.41·43-s + 1.40·47-s + 0.142·49-s + 1.14·53-s − 1.17·55-s + 1.04·59-s − 0.165·61-s − 0.402·67-s − 0.823·71-s − 0.234·73-s − 0.603·77-s + 0.820·79-s − 0.102·83-s + 1.00·85-s − 0.772·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.525952422\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.525952422\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 1.64T + 5T^{2} \) |
| 11 | \( 1 - 5.29T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 5.64T + 17T^{2} \) |
| 23 | \( 1 - 5.29T + 23T^{2} \) |
| 29 | \( 1 - 4.35T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 5.29T + 37T^{2} \) |
| 41 | \( 1 + 7.29T + 41T^{2} \) |
| 43 | \( 1 - 9.29T + 43T^{2} \) |
| 47 | \( 1 - 9.64T + 47T^{2} \) |
| 53 | \( 1 - 8.35T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 1.29T + 61T^{2} \) |
| 67 | \( 1 + 3.29T + 67T^{2} \) |
| 71 | \( 1 + 6.93T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 7.29T + 79T^{2} \) |
| 83 | \( 1 + 0.937T + 83T^{2} \) |
| 89 | \( 1 + 7.29T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
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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
−8.508961923463231501674625371977, −7.27574074168221298503454521521, −7.00949586396019603865287813349, −6.24423250463898592017295345507, −5.35761649033743834776372244054, −4.29602087204456176418457931846, −3.91397625368970238504918286116, −3.00781247064544217646624377372, −1.87369371235495819777570242150, −0.68526133251869615826265961193,
0.68526133251869615826265961193, 1.87369371235495819777570242150, 3.00781247064544217646624377372, 3.91397625368970238504918286116, 4.29602087204456176418457931846, 5.35761649033743834776372244054, 6.24423250463898592017295345507, 7.00949586396019603865287813349, 7.27574074168221298503454521521, 8.508961923463231501674625371977
