| L(s) = 1 | − 3-s + 3.23·5-s − 2·7-s + 9-s − 3.23·11-s − 4.47·13-s − 3.23·15-s − 6.47·17-s + 1.23·19-s + 2·21-s − 23-s + 5.47·25-s − 27-s − 8.47·29-s − 1.52·31-s + 3.23·33-s − 6.47·35-s + 7.70·37-s + 4.47·39-s − 2·41-s − 5.23·43-s + 3.23·45-s + 8.94·47-s − 3·49-s + 6.47·51-s + 12.1·53-s − 10.4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.44·5-s − 0.755·7-s + 0.333·9-s − 0.975·11-s − 1.24·13-s − 0.835·15-s − 1.56·17-s + 0.283·19-s + 0.436·21-s − 0.208·23-s + 1.09·25-s − 0.192·27-s − 1.57·29-s − 0.274·31-s + 0.563·33-s − 1.09·35-s + 1.26·37-s + 0.716·39-s − 0.312·41-s − 0.798·43-s + 0.482·45-s + 1.30·47-s − 0.428·49-s + 0.906·51-s + 1.67·53-s − 1.41·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 3.23T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 - 1.23T + 19T^{2} \) |
| 29 | \( 1 + 8.47T + 29T^{2} \) |
| 31 | \( 1 + 1.52T + 31T^{2} \) |
| 37 | \( 1 - 7.70T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 5.23T + 43T^{2} \) |
| 47 | \( 1 - 8.94T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 7.70T + 67T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 - 4.47T + 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 - 4.47T + 97T^{2} \) |
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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
−9.591942288284233575214493745773, −8.936858967277850470631409857428, −7.54425134821613058845972474817, −6.81555556957440783625848279071, −5.91484851724357785993658858451, −5.36445552464242304197929353756, −4.37421207907507972455177860046, −2.76804928053194146162740361594, −1.98461885292268035081600019357, 0,
1.98461885292268035081600019357, 2.76804928053194146162740361594, 4.37421207907507972455177860046, 5.36445552464242304197929353756, 5.91484851724357785993658858451, 6.81555556957440783625848279071, 7.54425134821613058845972474817, 8.936858967277850470631409857428, 9.591942288284233575214493745773
