| L(s) = 1 | − 0.814·2-s + 2.50·3-s − 1.33·4-s − 2.03·6-s + 4.83·7-s + 2.71·8-s + 3.25·9-s − 11-s − 3.34·12-s + 4.12·13-s − 3.94·14-s + 0.459·16-s − 2.02·17-s − 2.64·18-s + 19-s + 12.0·21-s + 0.814·22-s − 6.38·23-s + 6.79·24-s − 3.35·26-s + 0.626·27-s − 6.46·28-s + 7.42·29-s − 4.40·31-s − 5.80·32-s − 2.50·33-s + 1.65·34-s + ⋯ |
| L(s) = 1 | − 0.575·2-s + 1.44·3-s − 0.668·4-s − 0.831·6-s + 1.82·7-s + 0.960·8-s + 1.08·9-s − 0.301·11-s − 0.964·12-s + 1.14·13-s − 1.05·14-s + 0.114·16-s − 0.491·17-s − 0.624·18-s + 0.229·19-s + 2.64·21-s + 0.173·22-s − 1.33·23-s + 1.38·24-s − 0.658·26-s + 0.120·27-s − 1.22·28-s + 1.37·29-s − 0.791·31-s − 1.02·32-s − 0.435·33-s + 0.283·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.829355043\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.829355043\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 0.814T + 2T^{2} \) |
| 3 | \( 1 - 2.50T + 3T^{2} \) |
| 7 | \( 1 - 4.83T + 7T^{2} \) |
| 13 | \( 1 - 4.12T + 13T^{2} \) |
| 17 | \( 1 + 2.02T + 17T^{2} \) |
| 23 | \( 1 + 6.38T + 23T^{2} \) |
| 29 | \( 1 - 7.42T + 29T^{2} \) |
| 31 | \( 1 + 4.40T + 31T^{2} \) |
| 37 | \( 1 + 1.77T + 37T^{2} \) |
| 41 | \( 1 - 9.92T + 41T^{2} \) |
| 43 | \( 1 - 9.89T + 43T^{2} \) |
| 47 | \( 1 + 0.145T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 - 1.17T + 59T^{2} \) |
| 61 | \( 1 - 2.03T + 61T^{2} \) |
| 67 | \( 1 + 7.84T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 3.27T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 2.37T + 83T^{2} \) |
| 89 | \( 1 + 9.26T + 89T^{2} \) |
| 97 | \( 1 + 8.29T + 97T^{2} \) |
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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
−8.389559620063854657158146603372, −7.73397697735631548680163721061, −7.39941526449405982883314854952, −5.97650476619527079419425321649, −5.14988507173245207051569318239, −4.16830971656777077253310052524, −3.98210941745065109065857284253, −2.61785980887038576160409116741, −1.84200598797527007065576960172, −1.01813688135207007702235205849,
1.01813688135207007702235205849, 1.84200598797527007065576960172, 2.61785980887038576160409116741, 3.98210941745065109065857284253, 4.16830971656777077253310052524, 5.14988507173245207051569318239, 5.97650476619527079419425321649, 7.39941526449405982883314854952, 7.73397697735631548680163721061, 8.389559620063854657158146603372
