L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s + 2·13-s + 14-s + 16-s + 6·17-s + 8·19-s − 20-s + 25-s + 2·26-s + 28-s − 6·29-s − 4·31-s + 32-s + 6·34-s − 35-s − 10·37-s + 8·38-s − 40-s + 6·41-s − 4·43-s + 49-s + 50-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 1.83·19-s − 0.223·20-s + 1/5·25-s + 0.392·26-s + 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 1.02·34-s − 0.169·35-s − 1.64·37-s + 1.29·38-s − 0.158·40-s + 0.937·41-s − 0.609·43-s + 1/7·49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.334574167\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.334574167\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−10.77509849140855743529390892850, −9.857402504230161582528854630050, −8.790922026040486061843055845413, −7.67845508591579519153392227466, −7.19801386358340821477645347390, −5.76093596813527562032392850894, −5.20272031164140682979668587117, −3.88273199773070623460496754838, −3.13125557587754062819456448397, −1.41006155845401754086894809680,
1.41006155845401754086894809680, 3.13125557587754062819456448397, 3.88273199773070623460496754838, 5.20272031164140682979668587117, 5.76093596813527562032392850894, 7.19801386358340821477645347390, 7.67845508591579519153392227466, 8.790922026040486061843055845413, 9.857402504230161582528854630050, 10.77509849140855743529390892850