| L(s) = 1 | − 1.61·3-s − 3.23·5-s − 0.236·7-s − 0.381·9-s − 4.38·11-s + 3.47·13-s + 5.23·15-s + 7.23·17-s + 0.381·21-s + 5.85·23-s + 5.47·25-s + 5.47·27-s − 2.85·29-s − 4.38·31-s + 7.09·33-s + 0.763·35-s − 6.38·37-s − 5.61·39-s + 7.94·41-s − 0.618·43-s + 1.23·45-s + 12.7·47-s − 6.94·49-s − 11.7·51-s + 3.85·53-s + 14.1·55-s − 5.38·59-s + ⋯ |
| L(s) = 1 | − 0.934·3-s − 1.44·5-s − 0.0892·7-s − 0.127·9-s − 1.32·11-s + 0.962·13-s + 1.35·15-s + 1.75·17-s + 0.0833·21-s + 1.22·23-s + 1.09·25-s + 1.05·27-s − 0.529·29-s − 0.787·31-s + 1.23·33-s + 0.129·35-s − 1.04·37-s − 0.899·39-s + 1.24·41-s − 0.0942·43-s + 0.184·45-s + 1.85·47-s − 0.992·49-s − 1.63·51-s + 0.529·53-s + 1.91·55-s − 0.700·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 1.61T + 3T^{2} \) |
| 5 | \( 1 + 3.23T + 5T^{2} \) |
| 7 | \( 1 + 0.236T + 7T^{2} \) |
| 11 | \( 1 + 4.38T + 11T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 - 7.23T + 17T^{2} \) |
| 23 | \( 1 - 5.85T + 23T^{2} \) |
| 29 | \( 1 + 2.85T + 29T^{2} \) |
| 31 | \( 1 + 4.38T + 31T^{2} \) |
| 37 | \( 1 + 6.38T + 37T^{2} \) |
| 41 | \( 1 - 7.94T + 41T^{2} \) |
| 43 | \( 1 + 0.618T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 - 3.85T + 53T^{2} \) |
| 59 | \( 1 + 5.38T + 59T^{2} \) |
| 61 | \( 1 - 6.70T + 61T^{2} \) |
| 67 | \( 1 + 6.70T + 67T^{2} \) |
| 71 | \( 1 + 15.1T + 71T^{2} \) |
| 73 | \( 1 - 7.76T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 + 14.9T + 83T^{2} \) |
| 89 | \( 1 + 6.70T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−8.296355417939402569053206991125, −7.54132845570956291937068176826, −7.09701682765787328155622551841, −5.78434605016454427490933402072, −5.49578618468851791228068597263, −4.53200239990548450795661561263, −3.55347555214083088586107200369, −2.90738973544027064075301529215, −1.06627340503600558928652438602, 0,
1.06627340503600558928652438602, 2.90738973544027064075301529215, 3.55347555214083088586107200369, 4.53200239990548450795661561263, 5.49578618468851791228068597263, 5.78434605016454427490933402072, 7.09701682765787328155622551841, 7.54132845570956291937068176826, 8.296355417939402569053206991125
