| 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
| \(L(1)\) |
\(\approx\) |
\(2.951800582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.951800582\) |
| \(L(\frac{3}{2})\) |
|
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
−7.942131793242153015542718965397, −7.00930101989920149591893346148, −6.48975303716553484039083711781, −5.39712160895559853900097800738, −5.17830042835693109757991130576, −4.30424991845818340997989067029, −3.67791941925558480263583574607, −3.01436607424833740306909198472, −1.79545310967620180218745117614, −0.797915353672243560754955172632,
0.797915353672243560754955172632, 1.79545310967620180218745117614, 3.01436607424833740306909198472, 3.67791941925558480263583574607, 4.30424991845818340997989067029, 5.17830042835693109757991130576, 5.39712160895559853900097800738, 6.48975303716553484039083711781, 7.00930101989920149591893346148, 7.942131793242153015542718965397
