L(s) = 1 | + 2.62·3-s − 3.04·5-s − 5.08·7-s + 3.90·9-s − 5.20·11-s − 2.31·13-s − 8.00·15-s + 4.27·17-s + 1.59·19-s − 13.3·21-s − 2.43·23-s + 4.26·25-s + 2.38·27-s + 4.31·29-s − 4.46·31-s − 13.6·33-s + 15.4·35-s − 4.67·37-s − 6.07·39-s − 6.47·41-s + 0.211·43-s − 11.8·45-s + 11.7·47-s + 18.8·49-s + 11.2·51-s + 1.30·53-s + 15.8·55-s + ⋯ |
L(s) = 1 | + 1.51·3-s − 1.36·5-s − 1.92·7-s + 1.30·9-s − 1.56·11-s − 0.641·13-s − 2.06·15-s + 1.03·17-s + 0.366·19-s − 2.91·21-s − 0.508·23-s + 0.853·25-s + 0.458·27-s + 0.802·29-s − 0.801·31-s − 2.38·33-s + 2.61·35-s − 0.768·37-s − 0.972·39-s − 1.01·41-s + 0.0323·43-s − 1.77·45-s + 1.72·47-s + 2.69·49-s + 1.57·51-s + 0.179·53-s + 2.13·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 167 | \( 1 - T \) |
good | 3 | \( 1 - 2.62T + 3T^{2} \) |
| 5 | \( 1 + 3.04T + 5T^{2} \) |
| 7 | \( 1 + 5.08T + 7T^{2} \) |
| 11 | \( 1 + 5.20T + 11T^{2} \) |
| 13 | \( 1 + 2.31T + 13T^{2} \) |
| 17 | \( 1 - 4.27T + 17T^{2} \) |
| 19 | \( 1 - 1.59T + 19T^{2} \) |
| 23 | \( 1 + 2.43T + 23T^{2} \) |
| 29 | \( 1 - 4.31T + 29T^{2} \) |
| 31 | \( 1 + 4.46T + 31T^{2} \) |
| 37 | \( 1 + 4.67T + 37T^{2} \) |
| 41 | \( 1 + 6.47T + 41T^{2} \) |
| 43 | \( 1 - 0.211T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 - 1.30T + 53T^{2} \) |
| 59 | \( 1 + 2.08T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 - 15.7T + 73T^{2} \) |
| 79 | \( 1 + 1.89T + 79T^{2} \) |
| 83 | \( 1 - 0.0733T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 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
−9.940212985706159638639707408157, −9.201675712318600585992356814478, −8.203293210019771373929442592153, −7.62642177735570129178937467953, −6.96367210118076571635112699104, −5.47165941210579774926995384383, −3.99127445710170356155174144064, −3.23384300740786026510654415496, −2.66927766585803051681785782558, 0,
2.66927766585803051681785782558, 3.23384300740786026510654415496, 3.99127445710170356155174144064, 5.47165941210579774926995384383, 6.96367210118076571635112699104, 7.62642177735570129178937467953, 8.203293210019771373929442592153, 9.201675712318600585992356814478, 9.940212985706159638639707408157