| L(s) = 1 | + 1.56·2-s − 3-s + 0.438·4-s − 0.561·5-s − 1.56·6-s − 2.43·8-s + 9-s − 0.876·10-s − 2.56·11-s − 0.438·12-s + 4.56·13-s + 0.561·15-s − 4.68·16-s + 17-s + 1.56·18-s + 7.68·19-s − 0.246·20-s − 4·22-s − 6.56·23-s + 2.43·24-s − 4.68·25-s + 7.12·26-s − 27-s + 8.24·29-s + 0.876·30-s − 5.12·31-s − 2.43·32-s + ⋯ |
| L(s) = 1 | + 1.10·2-s − 0.577·3-s + 0.219·4-s − 0.251·5-s − 0.637·6-s − 0.862·8-s + 0.333·9-s − 0.277·10-s − 0.772·11-s − 0.126·12-s + 1.26·13-s + 0.144·15-s − 1.17·16-s + 0.242·17-s + 0.368·18-s + 1.76·19-s − 0.0550·20-s − 0.852·22-s − 1.36·23-s + 0.497·24-s − 0.936·25-s + 1.39·26-s − 0.192·27-s + 1.53·29-s + 0.160·30-s − 0.920·31-s − 0.431·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.051442404\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.051442404\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - 1.56T + 2T^{2} \) |
| 5 | \( 1 + 0.561T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 13 | \( 1 - 4.56T + 13T^{2} \) |
| 19 | \( 1 - 7.68T + 19T^{2} \) |
| 23 | \( 1 + 6.56T + 23T^{2} \) |
| 29 | \( 1 - 8.24T + 29T^{2} \) |
| 31 | \( 1 + 5.12T + 31T^{2} \) |
| 37 | \( 1 - 3.12T + 37T^{2} \) |
| 41 | \( 1 - 0.561T + 41T^{2} \) |
| 43 | \( 1 + 7.68T + 43T^{2} \) |
| 47 | \( 1 + 2.87T + 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 + 1.12T + 59T^{2} \) |
| 61 | \( 1 - 0.876T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 4.24T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 + 9.12T + 83T^{2} \) |
| 89 | \( 1 - 7.12T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−15.63807158947086580164143629245, −14.12855321780393269066450680309, −13.35475106935025158557411030500, −12.19823223663652320369476777140, −11.31545414338490342330639168297, −9.795998426494042820793301140912, −8.041139099955784602251453627133, −6.22365576022438585877086682446, −5.12309176107160292787145742500, −3.57400879965649271350359775461,
3.57400879965649271350359775461, 5.12309176107160292787145742500, 6.22365576022438585877086682446, 8.041139099955784602251453627133, 9.795998426494042820793301140912, 11.31545414338490342330639168297, 12.19823223663652320369476777140, 13.35475106935025158557411030500, 14.12855321780393269066450680309, 15.63807158947086580164143629245
