| L(s) = 1 | − 1.13·2-s − 0.714·4-s + 0.407·5-s + 2.15·7-s + 3.07·8-s − 0.461·10-s − 3.55·11-s − 0.0966·13-s − 2.44·14-s − 2.05·16-s − 0.587·17-s + 3.61·19-s − 0.291·20-s + 4.02·22-s − 8.26·23-s − 4.83·25-s + 0.109·26-s − 1.54·28-s − 1.67·31-s − 3.82·32-s + 0.665·34-s + 0.878·35-s + 0.0122·37-s − 4.10·38-s + 1.25·40-s + 9.57·41-s − 3.47·43-s + ⋯ |
| L(s) = 1 | − 0.801·2-s − 0.357·4-s + 0.182·5-s + 0.815·7-s + 1.08·8-s − 0.145·10-s − 1.07·11-s − 0.0268·13-s − 0.653·14-s − 0.514·16-s − 0.142·17-s + 0.830·19-s − 0.0650·20-s + 0.858·22-s − 1.72·23-s − 0.966·25-s + 0.0214·26-s − 0.291·28-s − 0.301·31-s − 0.675·32-s + 0.114·34-s + 0.148·35-s + 0.00202·37-s − 0.665·38-s + 0.198·40-s + 1.49·41-s − 0.530·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9162989105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9162989105\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 1.13T + 2T^{2} \) |
| 5 | \( 1 - 0.407T + 5T^{2} \) |
| 7 | \( 1 - 2.15T + 7T^{2} \) |
| 11 | \( 1 + 3.55T + 11T^{2} \) |
| 13 | \( 1 + 0.0966T + 13T^{2} \) |
| 17 | \( 1 + 0.587T + 17T^{2} \) |
| 19 | \( 1 - 3.61T + 19T^{2} \) |
| 23 | \( 1 + 8.26T + 23T^{2} \) |
| 31 | \( 1 + 1.67T + 31T^{2} \) |
| 37 | \( 1 - 0.0122T + 37T^{2} \) |
| 41 | \( 1 - 9.57T + 41T^{2} \) |
| 43 | \( 1 + 3.47T + 43T^{2} \) |
| 47 | \( 1 + 3.31T + 47T^{2} \) |
| 53 | \( 1 + 6.88T + 53T^{2} \) |
| 59 | \( 1 - 6.45T + 59T^{2} \) |
| 61 | \( 1 - 9.67T + 61T^{2} \) |
| 67 | \( 1 - 8.02T + 67T^{2} \) |
| 71 | \( 1 - 9.30T + 71T^{2} \) |
| 73 | \( 1 - 9.28T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 2.23T + 89T^{2} \) |
| 97 | \( 1 + 5.72T + 97T^{2} \) |
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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
−7.962804554713970590171740883939, −7.62527128958780248315524806817, −6.63273018605150687646677304493, −5.60233745632645068824528821067, −5.17857814466151424707271668323, −4.35810318471914091253424672550, −3.62992164238184430924244322196, −2.35267056768649447646175877615, −1.70924272134575210928263780891, −0.54797940048731320516837461728,
0.54797940048731320516837461728, 1.70924272134575210928263780891, 2.35267056768649447646175877615, 3.62992164238184430924244322196, 4.35810318471914091253424672550, 5.17857814466151424707271668323, 5.60233745632645068824528821067, 6.63273018605150687646677304493, 7.62527128958780248315524806817, 7.962804554713970590171740883939
