| L(s) = 1 | + 2-s − 1.40·3-s + 4-s + 2.72·5-s − 1.40·6-s − 7-s + 8-s − 1.02·9-s + 2.72·10-s + 2.81·11-s − 1.40·12-s + 4.13·13-s − 14-s − 3.83·15-s + 16-s − 7.12·17-s − 1.02·18-s − 5.00·19-s + 2.72·20-s + 1.40·21-s + 2.81·22-s − 7.67·23-s − 1.40·24-s + 2.44·25-s + 4.13·26-s + 5.65·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.811·3-s + 0.5·4-s + 1.22·5-s − 0.573·6-s − 0.377·7-s + 0.353·8-s − 0.341·9-s + 0.862·10-s + 0.848·11-s − 0.405·12-s + 1.14·13-s − 0.267·14-s − 0.989·15-s + 0.250·16-s − 1.72·17-s − 0.241·18-s − 1.14·19-s + 0.610·20-s + 0.306·21-s + 0.599·22-s − 1.59·23-s − 0.286·24-s + 0.488·25-s + 0.811·26-s + 1.08·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.825346477\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.825346477\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 1.40T + 3T^{2} \) |
| 5 | \( 1 - 2.72T + 5T^{2} \) |
| 11 | \( 1 - 2.81T + 11T^{2} \) |
| 13 | \( 1 - 4.13T + 13T^{2} \) |
| 17 | \( 1 + 7.12T + 17T^{2} \) |
| 19 | \( 1 + 5.00T + 19T^{2} \) |
| 23 | \( 1 + 7.67T + 23T^{2} \) |
| 29 | \( 1 - 4.14T + 29T^{2} \) |
| 31 | \( 1 - 7.50T + 31T^{2} \) |
| 37 | \( 1 - 5.81T + 37T^{2} \) |
| 41 | \( 1 - 6.11T + 41T^{2} \) |
| 43 | \( 1 - 5.49T + 43T^{2} \) |
| 47 | \( 1 + 5.67T + 47T^{2} \) |
| 53 | \( 1 - 5.80T + 53T^{2} \) |
| 59 | \( 1 - 9.67T + 59T^{2} \) |
| 61 | \( 1 - 14.0T + 61T^{2} \) |
| 67 | \( 1 + 5.47T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 + 6.15T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 - 8.75T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−8.247216149949658307272108207519, −6.67246508900138513910488237056, −6.46975321486282033782738038408, −6.06352609453099214692643398075, −5.42824487295411663608372212928, −4.35720055426934950850051733624, −3.97711932953500073959997519345, −2.60748325350133197186002182173, −2.06155389724253602349352463521, −0.823675309830865055051334982321,
0.823675309830865055051334982321, 2.06155389724253602349352463521, 2.60748325350133197186002182173, 3.97711932953500073959997519345, 4.35720055426934950850051733624, 5.42824487295411663608372212928, 6.06352609453099214692643398075, 6.46975321486282033782738038408, 6.67246508900138513910488237056, 8.247216149949658307272108207519
