| L(s) = 1 | + 3-s + 1.63·5-s − 3·7-s + 9-s + 4.34·11-s − 0.921·13-s + 1.63·15-s − 5.04·17-s − 0.630·19-s − 3·21-s − 1.63·23-s − 2.34·25-s + 27-s − 6.97·29-s − 6.41·31-s + 4.34·33-s − 4.89·35-s + 2.97·37-s − 0.921·39-s + 6.12·41-s + 1.21·43-s + 1.63·45-s − 5.75·47-s + 2·49-s − 5.04·51-s + 0.0494·53-s + 7.07·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.729·5-s − 1.13·7-s + 0.333·9-s + 1.30·11-s − 0.255·13-s + 0.421·15-s − 1.22·17-s − 0.144·19-s − 0.654·21-s − 0.340·23-s − 0.468·25-s + 0.192·27-s − 1.29·29-s − 1.15·31-s + 0.755·33-s − 0.827·35-s + 0.488·37-s − 0.147·39-s + 0.957·41-s + 0.184·43-s + 0.243·45-s − 0.839·47-s + 0.285·49-s − 0.707·51-s + 0.00679·53-s + 0.954·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 251 | \( 1 - T \) |
| good | 5 | \( 1 - 1.63T + 5T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 4.34T + 11T^{2} \) |
| 13 | \( 1 + 0.921T + 13T^{2} \) |
| 17 | \( 1 + 5.04T + 17T^{2} \) |
| 19 | \( 1 + 0.630T + 19T^{2} \) |
| 23 | \( 1 + 1.63T + 23T^{2} \) |
| 29 | \( 1 + 6.97T + 29T^{2} \) |
| 31 | \( 1 + 6.41T + 31T^{2} \) |
| 37 | \( 1 - 2.97T + 37T^{2} \) |
| 41 | \( 1 - 6.12T + 41T^{2} \) |
| 43 | \( 1 - 1.21T + 43T^{2} \) |
| 47 | \( 1 + 5.75T + 47T^{2} \) |
| 53 | \( 1 - 0.0494T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 + 6.04T + 61T^{2} \) |
| 67 | \( 1 - 9.65T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 + 14.5T + 73T^{2} \) |
| 79 | \( 1 - 3.34T + 79T^{2} \) |
| 83 | \( 1 + 7.04T + 83T^{2} \) |
| 89 | \( 1 - 1.23T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−7.57546974278497418113413795660, −7.03490470176177087601293719771, −6.13899478617991310067054971727, −5.98692400305694581564979933435, −4.65965882107925816886746698211, −3.93092161204401181361621496586, −3.25368249435778958234583271249, −2.26734792171745550166660010790, −1.58415690006203426646410814218, 0,
1.58415690006203426646410814218, 2.26734792171745550166660010790, 3.25368249435778958234583271249, 3.93092161204401181361621496586, 4.65965882107925816886746698211, 5.98692400305694581564979933435, 6.13899478617991310067054971727, 7.03490470176177087601293719771, 7.57546974278497418113413795660
