| L(s) = 1 | + 0.817·2-s + 1.89·3-s − 1.33·4-s + 1.54·6-s + 7-s − 2.72·8-s + 0.591·9-s + 2.12·11-s − 2.52·12-s + 5.55·13-s + 0.817·14-s + 0.436·16-s − 0.145·17-s + 0.483·18-s − 3.23·19-s + 1.89·21-s + 1.74·22-s + 23-s − 5.16·24-s + 4.53·26-s − 4.56·27-s − 1.33·28-s + 3.26·29-s + 0.223·31-s + 5.80·32-s + 4.03·33-s − 0.118·34-s + ⋯ |
| L(s) = 1 | + 0.578·2-s + 1.09·3-s − 0.665·4-s + 0.632·6-s + 0.377·7-s − 0.963·8-s + 0.197·9-s + 0.641·11-s − 0.728·12-s + 1.53·13-s + 0.218·14-s + 0.109·16-s − 0.0352·17-s + 0.113·18-s − 0.741·19-s + 0.413·21-s + 0.371·22-s + 0.208·23-s − 1.05·24-s + 0.890·26-s − 0.878·27-s − 0.251·28-s + 0.607·29-s + 0.0401·31-s + 1.02·32-s + 0.702·33-s − 0.0203·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.434570443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.434570443\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 0.817T + 2T^{2} \) |
| 3 | \( 1 - 1.89T + 3T^{2} \) |
| 11 | \( 1 - 2.12T + 11T^{2} \) |
| 13 | \( 1 - 5.55T + 13T^{2} \) |
| 17 | \( 1 + 0.145T + 17T^{2} \) |
| 19 | \( 1 + 3.23T + 19T^{2} \) |
| 29 | \( 1 - 3.26T + 29T^{2} \) |
| 31 | \( 1 - 0.223T + 31T^{2} \) |
| 37 | \( 1 + 1.16T + 37T^{2} \) |
| 41 | \( 1 + 2.79T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 - 2.13T + 47T^{2} \) |
| 53 | \( 1 - 7.28T + 53T^{2} \) |
| 59 | \( 1 - 4.72T + 59T^{2} \) |
| 61 | \( 1 - 2.99T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 - 1.35T + 71T^{2} \) |
| 73 | \( 1 - 5.64T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 + 16.1T + 83T^{2} \) |
| 89 | \( 1 - 5.37T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−8.620919980185399522313662398313, −8.023411397773694279528961378683, −6.94730375162414730789693115414, −6.08568903663364551025914830465, −5.46556972515990600697952154557, −4.33620343872245959860899817590, −3.89512423017060563627889961098, −3.17658326060595762895333059673, −2.20685399286747081386876903990, −0.969190644660726352556262536853,
0.969190644660726352556262536853, 2.20685399286747081386876903990, 3.17658326060595762895333059673, 3.89512423017060563627889961098, 4.33620343872245959860899817590, 5.46556972515990600697952154557, 6.08568903663364551025914830465, 6.94730375162414730789693115414, 8.023411397773694279528961378683, 8.620919980185399522313662398313
