L(s) = 1 | − 5-s − 2.82·7-s − 3·9-s + 4·11-s − 4.82·13-s + 7.65·17-s − 19-s − 2.82·23-s + 25-s + 3.65·29-s + 5.65·31-s + 2.82·35-s + 6.48·37-s − 3.65·41-s + 8.48·43-s + 3·45-s + 5.17·47-s + 1.00·49-s − 7.17·53-s − 4·55-s − 9.65·59-s − 6·61-s + 8.48·63-s + 4.82·65-s − 11.3·67-s + 5.65·71-s + 15.6·73-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 1.06·7-s − 9-s + 1.20·11-s − 1.33·13-s + 1.85·17-s − 0.229·19-s − 0.589·23-s + 0.200·25-s + 0.679·29-s + 1.01·31-s + 0.478·35-s + 1.06·37-s − 0.571·41-s + 1.29·43-s + 0.447·45-s + 0.754·47-s + 0.142·49-s − 0.985·53-s − 0.539·55-s − 1.25·59-s − 0.768·61-s + 1.06·63-s + 0.598·65-s − 1.38·67-s + 0.671·71-s + 1.83·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + 3T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 - 7.65T + 17T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 - 3.65T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 - 6.48T + 37T^{2} \) |
| 41 | \( 1 + 3.65T + 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 - 5.17T + 47T^{2} \) |
| 53 | \( 1 + 7.17T + 53T^{2} \) |
| 59 | \( 1 + 9.65T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 + 5.65T + 79T^{2} \) |
| 83 | \( 1 + 5.17T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 - 7.17T + 97T^{2} \) |
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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
−7.79326041691451744824114874533, −6.98216089460379869753799510756, −6.24404765592405336953248071773, −5.76066753189190162578981915101, −4.76652703118763336073145915062, −3.96388883699780316487747790824, −3.12753437923438136331575289245, −2.63197109039988661671267546503, −1.13699710816821334803465278597, 0,
1.13699710816821334803465278597, 2.63197109039988661671267546503, 3.12753437923438136331575289245, 3.96388883699780316487747790824, 4.76652703118763336073145915062, 5.76066753189190162578981915101, 6.24404765592405336953248071773, 6.98216089460379869753799510756, 7.79326041691451744824114874533