| L(s) = 1 | + 2.12·2-s + 3-s + 2.50·4-s + 2.12·6-s + 4.35·7-s + 1.07·8-s + 9-s − 1.57·11-s + 2.50·12-s + 1.19·13-s + 9.24·14-s − 2.73·16-s − 1.12·17-s + 2.12·18-s + 7.67·19-s + 4.35·21-s − 3.35·22-s − 2.32·23-s + 1.07·24-s + 2.54·26-s + 27-s + 10.9·28-s − 5.50·29-s + 4.80·31-s − 7.94·32-s − 1.57·33-s − 2.38·34-s + ⋯ |
| L(s) = 1 | + 1.50·2-s + 0.577·3-s + 1.25·4-s + 0.866·6-s + 1.64·7-s + 0.378·8-s + 0.333·9-s − 0.476·11-s + 0.722·12-s + 0.332·13-s + 2.47·14-s − 0.684·16-s − 0.272·17-s + 0.500·18-s + 1.76·19-s + 0.951·21-s − 0.714·22-s − 0.483·23-s + 0.218·24-s + 0.498·26-s + 0.192·27-s + 2.06·28-s − 1.02·29-s + 0.862·31-s − 1.40·32-s − 0.275·33-s − 0.408·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.597198879\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.597198879\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.12T + 2T^{2} \) |
| 7 | \( 1 - 4.35T + 7T^{2} \) |
| 11 | \( 1 + 1.57T + 11T^{2} \) |
| 13 | \( 1 - 1.19T + 13T^{2} \) |
| 17 | \( 1 + 1.12T + 17T^{2} \) |
| 19 | \( 1 - 7.67T + 19T^{2} \) |
| 23 | \( 1 + 2.32T + 23T^{2} \) |
| 29 | \( 1 + 5.50T + 29T^{2} \) |
| 31 | \( 1 - 4.80T + 31T^{2} \) |
| 37 | \( 1 + 6.37T + 37T^{2} \) |
| 41 | \( 1 + 7.47T + 41T^{2} \) |
| 43 | \( 1 + 1.24T + 43T^{2} \) |
| 47 | \( 1 - 4.12T + 47T^{2} \) |
| 53 | \( 1 - 3.74T + 53T^{2} \) |
| 59 | \( 1 - 9.15T + 59T^{2} \) |
| 61 | \( 1 - 1.39T + 61T^{2} \) |
| 67 | \( 1 + 3.86T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 - 4.99T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 + 8.73T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 + 7.49T + 97T^{2} \) |
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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
−9.074421989436409202087523112620, −8.290167918081240736115922005248, −7.58853283457500251845160639106, −6.81331167874248381779046024514, −5.52360225195437263871588884970, −5.19556974659761105682144448394, −4.30036636280699055946058168804, −3.51890248366556082225406699454, −2.52322699462522008965164135012, −1.53805828270281050196862763619,
1.53805828270281050196862763619, 2.52322699462522008965164135012, 3.51890248366556082225406699454, 4.30036636280699055946058168804, 5.19556974659761105682144448394, 5.52360225195437263871588884970, 6.81331167874248381779046024514, 7.58853283457500251845160639106, 8.290167918081240736115922005248, 9.074421989436409202087523112620
