| L(s) = 1 | + 5-s + 0.763·7-s − 3.23·11-s + 5.23·13-s + 4.47·17-s + 19-s + 6.47·23-s + 25-s − 5.23·29-s + 8.94·31-s + 0.763·35-s − 1.23·37-s − 3.70·41-s + 0.763·43-s + 6.47·47-s − 6.41·49-s − 8.47·53-s − 3.23·55-s + 1.52·59-s − 4.47·61-s + 5.23·65-s + 10.4·67-s + 3.52·73-s − 2.47·77-s − 4·83-s + 4.47·85-s − 11.7·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.288·7-s − 0.975·11-s + 1.45·13-s + 1.08·17-s + 0.229·19-s + 1.34·23-s + 0.200·25-s − 0.972·29-s + 1.60·31-s + 0.129·35-s − 0.203·37-s − 0.579·41-s + 0.116·43-s + 0.944·47-s − 0.916·49-s − 1.16·53-s − 0.436·55-s + 0.198·59-s − 0.572·61-s + 0.649·65-s + 1.27·67-s + 0.412·73-s − 0.281·77-s − 0.439·83-s + 0.485·85-s − 1.24·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.583233317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.583233317\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 0.763T + 7T^{2} \) |
| 11 | \( 1 + 3.23T + 11T^{2} \) |
| 13 | \( 1 - 5.23T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 23 | \( 1 - 6.47T + 23T^{2} \) |
| 29 | \( 1 + 5.23T + 29T^{2} \) |
| 31 | \( 1 - 8.94T + 31T^{2} \) |
| 37 | \( 1 + 1.23T + 37T^{2} \) |
| 41 | \( 1 + 3.70T + 41T^{2} \) |
| 43 | \( 1 - 0.763T + 43T^{2} \) |
| 47 | \( 1 - 6.47T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 - 1.52T + 59T^{2} \) |
| 61 | \( 1 + 4.47T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 3.52T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 + 9.23T + 97T^{2} \) |
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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
−8.124356408694151834427347085869, −7.28533704989434249056731873913, −6.53612594300121925932649227890, −5.73855621667658530456705381659, −5.27721586424447143641806594096, −4.46195137849900134905962177317, −3.40925684941585861492184765343, −2.85981373633926266754249138638, −1.70443795352673202194346302434, −0.874037518062871659142406326712,
0.874037518062871659142406326712, 1.70443795352673202194346302434, 2.85981373633926266754249138638, 3.40925684941585861492184765343, 4.46195137849900134905962177317, 5.27721586424447143641806594096, 5.73855621667658530456705381659, 6.53612594300121925932649227890, 7.28533704989434249056731873913, 8.124356408694151834427347085869
