| L(s) = 1 | + 2-s − 0.706·3-s + 4-s + 0.802·5-s − 0.706·6-s + 5.13·7-s + 8-s − 2.50·9-s + 0.802·10-s + 2.39·11-s − 0.706·12-s + 6.35·13-s + 5.13·14-s − 0.566·15-s + 16-s + 0.473·17-s − 2.50·18-s + 4.23·19-s + 0.802·20-s − 3.62·21-s + 2.39·22-s − 4.74·23-s − 0.706·24-s − 4.35·25-s + 6.35·26-s + 3.88·27-s + 5.13·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.407·3-s + 0.5·4-s + 0.358·5-s − 0.288·6-s + 1.94·7-s + 0.353·8-s − 0.833·9-s + 0.253·10-s + 0.720·11-s − 0.203·12-s + 1.76·13-s + 1.37·14-s − 0.146·15-s + 0.250·16-s + 0.114·17-s − 0.589·18-s + 0.972·19-s + 0.179·20-s − 0.791·21-s + 0.509·22-s − 0.989·23-s − 0.144·24-s − 0.871·25-s + 1.24·26-s + 0.747·27-s + 0.970·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.300186491\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.300186491\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 + T \) |
| good | 3 | \( 1 + 0.706T + 3T^{2} \) |
| 5 | \( 1 - 0.802T + 5T^{2} \) |
| 7 | \( 1 - 5.13T + 7T^{2} \) |
| 11 | \( 1 - 2.39T + 11T^{2} \) |
| 13 | \( 1 - 6.35T + 13T^{2} \) |
| 17 | \( 1 - 0.473T + 17T^{2} \) |
| 19 | \( 1 - 4.23T + 19T^{2} \) |
| 23 | \( 1 + 4.74T + 23T^{2} \) |
| 29 | \( 1 + 5.70T + 29T^{2} \) |
| 37 | \( 1 - 3.31T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 - 2.95T + 47T^{2} \) |
| 53 | \( 1 - 2.40T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 + 3.15T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 + 0.918T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 + 0.425T + 79T^{2} \) |
| 83 | \( 1 - 2.70T + 83T^{2} \) |
| 89 | \( 1 + 8.81T + 89T^{2} \) |
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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
−7.74084395538836408878077603489, −7.64894737444838151075218454336, −6.23961224408459108393065707005, −5.77589184681521456858073189659, −5.47093564680107115393960302137, −4.30652825363752506471079939669, −3.99317280379206973667629701454, −2.80179252002650189332291082016, −1.73877710082008533667355408721, −1.14080790659575774557075031080,
1.14080790659575774557075031080, 1.73877710082008533667355408721, 2.80179252002650189332291082016, 3.99317280379206973667629701454, 4.30652825363752506471079939669, 5.47093564680107115393960302137, 5.77589184681521456858073189659, 6.23961224408459108393065707005, 7.64894737444838151075218454336, 7.74084395538836408878077603489
