| L(s) = 1 | + 0.193·2-s + 3-s − 1.96·4-s + 5-s + 0.193·6-s − 3.35·7-s − 0.768·8-s + 9-s + 0.193·10-s − 1.96·12-s − 2.96·13-s − 0.649·14-s + 15-s + 3.77·16-s + 4.57·17-s + 0.193·18-s + 4.31·19-s − 1.96·20-s − 3.35·21-s − 6.70·23-s − 0.768·24-s + 25-s − 0.574·26-s + 27-s + 6.57·28-s + 3.61·29-s + 0.193·30-s + ⋯ |
| L(s) = 1 | + 0.137·2-s + 0.577·3-s − 0.981·4-s + 0.447·5-s + 0.0791·6-s − 1.26·7-s − 0.271·8-s + 0.333·9-s + 0.0613·10-s − 0.566·12-s − 0.821·13-s − 0.173·14-s + 0.258·15-s + 0.943·16-s + 1.10·17-s + 0.0457·18-s + 0.989·19-s − 0.438·20-s − 0.731·21-s − 1.39·23-s − 0.156·24-s + 0.200·25-s − 0.112·26-s + 0.192·27-s + 1.24·28-s + 0.670·29-s + 0.0354·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.612581653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.612581653\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.193T + 2T^{2} \) |
| 7 | \( 1 + 3.35T + 7T^{2} \) |
| 13 | \( 1 + 2.96T + 13T^{2} \) |
| 17 | \( 1 - 4.57T + 17T^{2} \) |
| 19 | \( 1 - 4.31T + 19T^{2} \) |
| 23 | \( 1 + 6.70T + 23T^{2} \) |
| 29 | \( 1 - 3.61T + 29T^{2} \) |
| 31 | \( 1 - 9.92T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 4.38T + 41T^{2} \) |
| 43 | \( 1 - 9.27T + 43T^{2} \) |
| 47 | \( 1 + 9.92T + 47T^{2} \) |
| 53 | \( 1 - 4.70T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 8.70T + 61T^{2} \) |
| 67 | \( 1 - 5.92T + 67T^{2} \) |
| 71 | \( 1 - 9.92T + 71T^{2} \) |
| 73 | \( 1 - 7.73T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 2.77T + 89T^{2} \) |
| 97 | \( 1 - 0.0752T + 97T^{2} \) |
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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
−9.576357678923802698672438082241, −8.493097193028742401718289954869, −7.87032407848544047831295562515, −6.88161925653178687353406807453, −5.96303005005814666030707535135, −5.21771802521927821732912709099, −4.17949852822205665608011889483, −3.33565697388660835294905882749, −2.54783851640045276144847827202, −0.832755459395975989292325862828,
0.832755459395975989292325862828, 2.54783851640045276144847827202, 3.33565697388660835294905882749, 4.17949852822205665608011889483, 5.21771802521927821732912709099, 5.96303005005814666030707535135, 6.88161925653178687353406807453, 7.87032407848544047831295562515, 8.493097193028742401718289954869, 9.576357678923802698672438082241
