L(s) = 1 | + 2.82·5-s − 7-s − 3·9-s − 1.41·11-s + 2·17-s + 2.82·19-s + 6·23-s + 3.00·25-s − 9.89·29-s − 8·31-s − 2.82·35-s − 7.07·37-s − 10·41-s + 1.41·43-s − 8.48·45-s + 12·47-s + 49-s + 1.41·53-s − 4.00·55-s + 11.3·59-s − 8.48·61-s + 3·63-s − 4.24·67-s + 6·73-s + 1.41·77-s − 10·79-s + 9·81-s + ⋯ |
L(s) = 1 | + 1.26·5-s − 0.377·7-s − 9-s − 0.426·11-s + 0.485·17-s + 0.648·19-s + 1.25·23-s + 0.600·25-s − 1.83·29-s − 1.43·31-s − 0.478·35-s − 1.16·37-s − 1.56·41-s + 0.215·43-s − 1.26·45-s + 1.75·47-s + 0.142·49-s + 0.194·53-s − 0.539·55-s + 1.47·59-s − 1.08·61-s + 0.377·63-s − 0.518·67-s + 0.702·73-s + 0.161·77-s − 1.12·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 - 2.82T + 5T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 9.89T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 7.07T + 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 1.41T + 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 - 1.41T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 8.48T + 61T^{2} \) |
| 67 | \( 1 + 4.24T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−7.35938796639628722880958806657, −6.99905868988859041639161215698, −5.88313191784736082030173388673, −5.55431636434913993935644867007, −5.12843098567273777149184630525, −3.74933445169574948604081685124, −3.07705762699578567791915645900, −2.29018960102591243275887472695, −1.41745580376626765700017176735, 0,
1.41745580376626765700017176735, 2.29018960102591243275887472695, 3.07705762699578567791915645900, 3.74933445169574948604081685124, 5.12843098567273777149184630525, 5.55431636434913993935644867007, 5.88313191784736082030173388673, 6.99905868988859041639161215698, 7.35938796639628722880958806657