| 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
−8.026446351643014671392956402712, −7.44996477294935955729009248197, −6.65578058895232185772423820533, −5.51888904533464177935143328873, −5.02340433996216459968722210003, −3.90719474335621605348449803691, −3.18403566940783483477247258835, −2.51813623389478717329283880583, −1.72126641835229459203742554724, 0,
1.72126641835229459203742554724, 2.51813623389478717329283880583, 3.18403566940783483477247258835, 3.90719474335621605348449803691, 5.02340433996216459968722210003, 5.51888904533464177935143328873, 6.65578058895232185772423820533, 7.44996477294935955729009248197, 8.026446351643014671392956402712
