| L(s) = 1 | + 1.32·2-s − 3-s − 0.236·4-s − 1.32·6-s − 2.96·8-s + 9-s + 4.47·11-s + 0.236·12-s − 0.763·13-s − 3.47·16-s − 5.23·17-s + 1.32·18-s + 1.64·19-s + 5.93·22-s − 4.29·23-s + 2.96·24-s − 1.01·26-s − 27-s + 2·29-s + 6.95·31-s + 1.32·32-s − 4.47·33-s − 6.95·34-s − 0.236·36-s + 8.59·37-s + 2.18·38-s + 0.763·39-s + ⋯ |
| L(s) = 1 | + 0.939·2-s − 0.577·3-s − 0.118·4-s − 0.542·6-s − 1.04·8-s + 0.333·9-s + 1.34·11-s + 0.0681·12-s − 0.211·13-s − 0.868·16-s − 1.26·17-s + 0.313·18-s + 0.376·19-s + 1.26·22-s − 0.896·23-s + 0.606·24-s − 0.198·26-s − 0.192·27-s + 0.371·29-s + 1.24·31-s + 0.234·32-s − 0.778·33-s − 1.19·34-s − 0.0393·36-s + 1.41·37-s + 0.353·38-s + 0.122·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 1.32T + 2T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 13 | \( 1 + 0.763T + 13T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 19 | \( 1 - 1.64T + 19T^{2} \) |
| 23 | \( 1 + 4.29T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 6.95T + 31T^{2} \) |
| 37 | \( 1 - 8.59T + 37T^{2} \) |
| 41 | \( 1 + 9.61T + 41T^{2} \) |
| 43 | \( 1 + 5.31T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 - 6.95T + 53T^{2} \) |
| 59 | \( 1 - 5.31T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 + 8.59T + 67T^{2} \) |
| 71 | \( 1 + 8.47T + 71T^{2} \) |
| 73 | \( 1 + 7.23T + 73T^{2} \) |
| 79 | \( 1 + 4.94T + 79T^{2} \) |
| 83 | \( 1 - 2.47T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−8.262600764232962195350089241523, −7.08305699273002086385116008993, −6.35442414216857315127656819905, −6.03383987214604929042484368408, −4.88676789102003341084631055008, −4.47469765176153258461175819243, −3.73449942060563684394729023893, −2.75229969270356912938948029589, −1.45924227046019157729297653497, 0,
1.45924227046019157729297653497, 2.75229969270356912938948029589, 3.73449942060563684394729023893, 4.47469765176153258461175819243, 4.88676789102003341084631055008, 6.03383987214604929042484368408, 6.35442414216857315127656819905, 7.08305699273002086385116008993, 8.262600764232962195350089241523
