| L(s) = 1 | + 2.61·2-s + 4.85·4-s + 5-s + 1.23·7-s + 7.47·8-s + 2.61·10-s + 3.23·11-s − 6.23·13-s + 3.23·14-s + 9.85·16-s − 2.47·17-s − 5.70·19-s + 4.85·20-s + 8.47·22-s + 23-s + 25-s − 16.3·26-s + 6.00·28-s + 0.527·29-s + 4.23·31-s + 10.8·32-s − 6.47·34-s + 1.23·35-s − 9.70·37-s − 14.9·38-s + 7.47·40-s + 7.47·41-s + ⋯ |
| L(s) = 1 | + 1.85·2-s + 2.42·4-s + 0.447·5-s + 0.467·7-s + 2.64·8-s + 0.827·10-s + 0.975·11-s − 1.72·13-s + 0.864·14-s + 2.46·16-s − 0.599·17-s − 1.30·19-s + 1.08·20-s + 1.80·22-s + 0.208·23-s + 0.200·25-s − 3.20·26-s + 1.13·28-s + 0.0980·29-s + 0.760·31-s + 1.91·32-s − 1.10·34-s + 0.208·35-s − 1.59·37-s − 2.42·38-s + 1.18·40-s + 1.16·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.303228837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.303228837\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 2.61T + 2T^{2} \) |
| 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 - 3.23T + 11T^{2} \) |
| 13 | \( 1 + 6.23T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 19 | \( 1 + 5.70T + 19T^{2} \) |
| 29 | \( 1 - 0.527T + 29T^{2} \) |
| 31 | \( 1 - 4.23T + 31T^{2} \) |
| 37 | \( 1 + 9.70T + 37T^{2} \) |
| 41 | \( 1 - 7.47T + 41T^{2} \) |
| 43 | \( 1 - 3.70T + 43T^{2} \) |
| 47 | \( 1 + 9.47T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 - 9.70T + 67T^{2} \) |
| 71 | \( 1 - 6.23T + 71T^{2} \) |
| 73 | \( 1 + 6.70T + 73T^{2} \) |
| 79 | \( 1 - 8.76T + 79T^{2} \) |
| 83 | \( 1 - 6.47T + 83T^{2} \) |
| 89 | \( 1 + 2.76T + 89T^{2} \) |
| 97 | \( 1 + 6.18T + 97T^{2} \) |
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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
−10.22230213966789659233711101689, −9.193635076160769045052030264163, −7.990866990432419619198521853498, −6.84992571650857851397512948085, −6.51178108616119320555298530601, −5.35327197858425933061173837693, −4.70127714638929670600053208422, −3.95826615543655299208560770771, −2.66088764077004621332134736837, −1.87973744359409733703097371343,
1.87973744359409733703097371343, 2.66088764077004621332134736837, 3.95826615543655299208560770771, 4.70127714638929670600053208422, 5.35327197858425933061173837693, 6.51178108616119320555298530601, 6.84992571650857851397512948085, 7.990866990432419619198521853498, 9.193635076160769045052030264163, 10.22230213966789659233711101689
