| L(s) = 1 | − 2.69·2-s − 2.62·3-s + 5.24·4-s + 0.394·5-s + 7.07·6-s − 0.479·7-s − 8.73·8-s + 3.90·9-s − 1.06·10-s − 1.84·11-s − 13.7·12-s + 6.41·13-s + 1.28·14-s − 1.03·15-s + 13.0·16-s − 0.224·17-s − 10.5·18-s + 2.59·19-s + 2.06·20-s + 1.25·21-s + 4.97·22-s + 3.01·23-s + 22.9·24-s − 4.84·25-s − 17.2·26-s − 2.37·27-s − 2.51·28-s + ⋯ |
| L(s) = 1 | − 1.90·2-s − 1.51·3-s + 2.62·4-s + 0.176·5-s + 2.88·6-s − 0.181·7-s − 3.08·8-s + 1.30·9-s − 0.335·10-s − 0.557·11-s − 3.97·12-s + 1.77·13-s + 0.344·14-s − 0.267·15-s + 3.25·16-s − 0.0545·17-s − 2.47·18-s + 0.596·19-s + 0.462·20-s + 0.274·21-s + 1.06·22-s + 0.628·23-s + 4.68·24-s − 0.968·25-s − 3.38·26-s − 0.456·27-s − 0.474·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + T \) |
| 163 | \( 1 + T \) |
| good | 2 | \( 1 + 2.69T + 2T^{2} \) |
| 3 | \( 1 + 2.62T + 3T^{2} \) |
| 5 | \( 1 - 0.394T + 5T^{2} \) |
| 7 | \( 1 + 0.479T + 7T^{2} \) |
| 11 | \( 1 + 1.84T + 11T^{2} \) |
| 13 | \( 1 - 6.41T + 13T^{2} \) |
| 17 | \( 1 + 0.224T + 17T^{2} \) |
| 19 | \( 1 - 2.59T + 19T^{2} \) |
| 23 | \( 1 - 3.01T + 23T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 + 0.0554T + 31T^{2} \) |
| 41 | \( 1 - 7.63T + 41T^{2} \) |
| 43 | \( 1 - 6.63T + 43T^{2} \) |
| 47 | \( 1 - 5.12T + 47T^{2} \) |
| 53 | \( 1 + 1.98T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 + 9.06T + 61T^{2} \) |
| 67 | \( 1 - 3.07T + 67T^{2} \) |
| 71 | \( 1 + 9.10T + 71T^{2} \) |
| 73 | \( 1 - 0.856T + 73T^{2} \) |
| 79 | \( 1 - 9.06T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−7.64823499226683035374786210744, −7.22466037955766264901733391744, −6.22533660951768668553241294733, −6.01008233480751589520595188378, −5.32472589655035683129143582586, −3.96248969548900484650328324404, −2.89210212480212459829302468109, −1.69031019653197519620120676305, −0.976379542065803451711349792396, 0,
0.976379542065803451711349792396, 1.69031019653197519620120676305, 2.89210212480212459829302468109, 3.96248969548900484650328324404, 5.32472589655035683129143582586, 6.01008233480751589520595188378, 6.22533660951768668553241294733, 7.22466037955766264901733391744, 7.64823499226683035374786210744
