| L(s) = 1 | − 2.27·3-s − 2.08·5-s − 1.11·7-s + 2.16·9-s − 4.08·11-s + 0.839·13-s + 4.74·15-s + 3.11·17-s − 2.97·19-s + 2.52·21-s + 23-s − 0.635·25-s + 1.90·27-s + 9.01·29-s + 0.315·31-s + 9.28·33-s + 2.32·35-s − 2.08·37-s − 1.90·39-s + 11.3·41-s − 0.478·43-s − 4.51·45-s + 8.22·47-s − 5.76·49-s − 7.06·51-s + 0.434·53-s + 8.54·55-s + ⋯ |
| L(s) = 1 | − 1.31·3-s − 0.934·5-s − 0.419·7-s + 0.720·9-s − 1.23·11-s + 0.232·13-s + 1.22·15-s + 0.754·17-s − 0.683·19-s + 0.550·21-s + 0.208·23-s − 0.127·25-s + 0.366·27-s + 1.67·29-s + 0.0566·31-s + 1.61·33-s + 0.392·35-s − 0.343·37-s − 0.305·39-s + 1.77·41-s − 0.0729·43-s − 0.672·45-s + 1.20·47-s − 0.823·49-s − 0.989·51-s + 0.0597·53-s + 1.15·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5395646715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5395646715\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 2.27T + 3T^{2} \) |
| 5 | \( 1 + 2.08T + 5T^{2} \) |
| 7 | \( 1 + 1.11T + 7T^{2} \) |
| 11 | \( 1 + 4.08T + 11T^{2} \) |
| 13 | \( 1 - 0.839T + 13T^{2} \) |
| 17 | \( 1 - 3.11T + 17T^{2} \) |
| 19 | \( 1 + 2.97T + 19T^{2} \) |
| 29 | \( 1 - 9.01T + 29T^{2} \) |
| 31 | \( 1 - 0.315T + 31T^{2} \) |
| 37 | \( 1 + 2.08T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 + 0.478T + 43T^{2} \) |
| 47 | \( 1 - 8.22T + 47T^{2} \) |
| 53 | \( 1 - 0.434T + 53T^{2} \) |
| 59 | \( 1 - 2.86T + 59T^{2} \) |
| 61 | \( 1 - 3.56T + 61T^{2} \) |
| 67 | \( 1 - 6.31T + 67T^{2} \) |
| 71 | \( 1 - 8.12T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + 0.454T + 83T^{2} \) |
| 89 | \( 1 - 4.32T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−10.64364040223946759881385778893, −9.817173172404929448948400274033, −8.475821889093788634402603964550, −7.74315049974501046828172284659, −6.76556360837859592719245708569, −5.88733770928463296846081361821, −5.06402164021602427031009734797, −4.08772964311413664848554504505, −2.79499550431469916594357888502, −0.63205016896101635829988777367,
0.63205016896101635829988777367, 2.79499550431469916594357888502, 4.08772964311413664848554504505, 5.06402164021602427031009734797, 5.88733770928463296846081361821, 6.76556360837859592719245708569, 7.74315049974501046828172284659, 8.475821889093788634402603964550, 9.817173172404929448948400274033, 10.64364040223946759881385778893
