| L(s) = 1 | − 2.28·3-s + 2.23·9-s − 5.47·11-s + 0.874·13-s − 4.57·17-s + 5.99·19-s + 3.47·23-s + 1.74·27-s + 0.236·29-s + 8.27·31-s + 12.5·33-s − 4.23·37-s − 2·39-s + 5.11·41-s − 3.76·43-s − 4.91·47-s + 10.4·51-s − 11.7·53-s − 13.7·57-s − 1.95·59-s + 7.53·61-s + 13.9·67-s − 7.94·69-s + 16.7·71-s + 7.53·73-s − 11.4·79-s − 10.7·81-s + ⋯ |
| L(s) = 1 | − 1.32·3-s + 0.745·9-s − 1.64·11-s + 0.242·13-s − 1.10·17-s + 1.37·19-s + 0.723·23-s + 0.336·27-s + 0.0438·29-s + 1.48·31-s + 2.17·33-s − 0.696·37-s − 0.320·39-s + 0.799·41-s − 0.573·43-s − 0.716·47-s + 1.46·51-s − 1.60·53-s − 1.81·57-s − 0.254·59-s + 0.964·61-s + 1.70·67-s − 0.956·69-s + 1.98·71-s + 0.881·73-s − 1.29·79-s − 1.18·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.28T + 3T^{2} \) |
| 11 | \( 1 + 5.47T + 11T^{2} \) |
| 13 | \( 1 - 0.874T + 13T^{2} \) |
| 17 | \( 1 + 4.57T + 17T^{2} \) |
| 19 | \( 1 - 5.99T + 19T^{2} \) |
| 23 | \( 1 - 3.47T + 23T^{2} \) |
| 29 | \( 1 - 0.236T + 29T^{2} \) |
| 31 | \( 1 - 8.27T + 31T^{2} \) |
| 37 | \( 1 + 4.23T + 37T^{2} \) |
| 41 | \( 1 - 5.11T + 41T^{2} \) |
| 43 | \( 1 + 3.76T + 43T^{2} \) |
| 47 | \( 1 + 4.91T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 + 1.95T + 59T^{2} \) |
| 61 | \( 1 - 7.53T + 61T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 - 16.7T + 71T^{2} \) |
| 73 | \( 1 - 7.53T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 5.86T + 89T^{2} \) |
| 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
−7.921888322923565788768593024514, −6.93882139217664179464800852402, −6.49458464771667452303232116857, −5.52193051087679007330810758480, −5.13797661824473811592355694251, −4.51337142485574537979034848896, −3.24641583269642046040499782575, −2.42959930948201715857015540002, −1.04878302878829698685967827180, 0,
1.04878302878829698685967827180, 2.42959930948201715857015540002, 3.24641583269642046040499782575, 4.51337142485574537979034848896, 5.13797661824473811592355694251, 5.52193051087679007330810758480, 6.49458464771667452303232116857, 6.93882139217664179464800852402, 7.921888322923565788768593024514
