L(s) = 1 | + 1.50·2-s + 0.256·4-s + 3.82·5-s + 7-s − 2.61·8-s + 5.74·10-s + 0.542·11-s − 1.21·13-s + 1.50·14-s − 4.44·16-s + 3.66·17-s + 5.00·19-s + 0.980·20-s + 0.814·22-s + 23-s + 9.62·25-s − 1.82·26-s + 0.256·28-s − 3.72·29-s + 9.04·31-s − 1.44·32-s + 5.50·34-s + 3.82·35-s − 9.08·37-s + 7.51·38-s − 10.0·40-s + 4.86·41-s + ⋯ |
L(s) = 1 | + 1.06·2-s + 0.128·4-s + 1.71·5-s + 0.377·7-s − 0.925·8-s + 1.81·10-s + 0.163·11-s − 0.337·13-s + 0.401·14-s − 1.11·16-s + 0.888·17-s + 1.14·19-s + 0.219·20-s + 0.173·22-s + 0.208·23-s + 1.92·25-s − 0.358·26-s + 0.0484·28-s − 0.692·29-s + 1.62·31-s − 0.254·32-s + 0.943·34-s + 0.646·35-s − 1.49·37-s + 1.21·38-s − 1.58·40-s + 0.759·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.692570126\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.692570126\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 - 1.50T + 2T^{2} \) |
| 5 | \( 1 - 3.82T + 5T^{2} \) |
| 11 | \( 1 - 0.542T + 11T^{2} \) |
| 13 | \( 1 + 1.21T + 13T^{2} \) |
| 17 | \( 1 - 3.66T + 17T^{2} \) |
| 19 | \( 1 - 5.00T + 19T^{2} \) |
| 29 | \( 1 + 3.72T + 29T^{2} \) |
| 31 | \( 1 - 9.04T + 31T^{2} \) |
| 37 | \( 1 + 9.08T + 37T^{2} \) |
| 41 | \( 1 - 4.86T + 41T^{2} \) |
| 43 | \( 1 - 3.64T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 7.48T + 53T^{2} \) |
| 59 | \( 1 + 5.18T + 59T^{2} \) |
| 61 | \( 1 - 3.26T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 - 5.29T + 71T^{2} \) |
| 73 | \( 1 - 8.19T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 + 7.64T + 83T^{2} \) |
| 89 | \( 1 - 4.74T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−9.596077409325054320846480451834, −8.959488451844814511053749396032, −7.85211150018347788038015815775, −6.70620096372441897678349123256, −5.97572633958835326963066484426, −5.28387919309721862778786081181, −4.78041150367619318643399276887, −3.43958663396641057493419974202, −2.57978377742495937801420634261, −1.36933971130617454660814374318,
1.36933971130617454660814374318, 2.57978377742495937801420634261, 3.43958663396641057493419974202, 4.78041150367619318643399276887, 5.28387919309721862778786081181, 5.97572633958835326963066484426, 6.70620096372441897678349123256, 7.85211150018347788038015815775, 8.959488451844814511053749396032, 9.596077409325054320846480451834