| L(s) = 1 | + 2-s + 2.82·3-s + 4-s − 0.0813·5-s + 2.82·6-s + 8-s + 4.99·9-s − 0.0813·10-s − 1.39·11-s + 2.82·12-s + 0.459·13-s − 0.229·15-s + 16-s + 4.67·17-s + 4.99·18-s − 2.88·19-s − 0.0813·20-s − 1.39·22-s + 7.66·23-s + 2.82·24-s − 4.99·25-s + 0.459·26-s + 5.63·27-s + 7.25·29-s − 0.229·30-s − 31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.63·3-s + 0.5·4-s − 0.0363·5-s + 1.15·6-s + 0.353·8-s + 1.66·9-s − 0.0257·10-s − 0.420·11-s + 0.816·12-s + 0.127·13-s − 0.0593·15-s + 0.250·16-s + 1.13·17-s + 1.17·18-s − 0.662·19-s − 0.0181·20-s − 0.297·22-s + 1.59·23-s + 0.577·24-s − 0.998·25-s + 0.0902·26-s + 1.08·27-s + 1.34·29-s − 0.0419·30-s − 0.179·31-s + 0.176·32-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
| \(L(1)\) |
\(\approx\) |
\(6.276944349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.276944349\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.82T + 3T^{2} \) |
| 5 | \( 1 + 0.0813T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 1.39T + 11T^{2} \) |
| 13 | \( 1 - 0.459T + 13T^{2} \) |
| 17 | \( 1 - 4.67T + 17T^{2} \) |
| 19 | \( 1 + 2.88T + 19T^{2} \) |
| 23 | \( 1 - 7.66T + 23T^{2} \) |
| 29 | \( 1 - 7.25T + 29T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 9.73T + 41T^{2} \) |
| 43 | \( 1 - 1.39T + 43T^{2} \) |
| 47 | \( 1 + 1.14T + 47T^{2} \) |
| 53 | \( 1 + 2.48T + 53T^{2} \) |
| 59 | \( 1 - 1.07T + 59T^{2} \) |
| 61 | \( 1 + 5.26T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 7.54T + 71T^{2} \) |
| 73 | \( 1 - 5.32T + 73T^{2} \) |
| 79 | \( 1 + 6.38T + 79T^{2} \) |
| 83 | \( 1 + 2.44T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
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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
−7.961318664285007819620601640124, −7.57278148651955456069410464896, −6.76070663956675002822663737462, −5.92982851581691000721534083334, −5.00703906612981027718661836156, −4.28403936755104808656212217846, −3.47252487525466862171088181316, −2.92305110468265889702433725155, −2.23103576259638108442341575767, −1.18469475860936936299925716691,
1.18469475860936936299925716691, 2.23103576259638108442341575767, 2.92305110468265889702433725155, 3.47252487525466862171088181316, 4.28403936755104808656212217846, 5.00703906612981027718661836156, 5.92982851581691000721534083334, 6.76070663956675002822663737462, 7.57278148651955456069410464896, 7.961318664285007819620601640124
