L(s) = 1 | + 3-s − 5-s − 1.64·7-s − 2·9-s − 1.64·11-s − 2.64·13-s − 15-s − 7.29·17-s − 5.64·19-s − 1.64·21-s + 23-s + 25-s − 5·27-s + 3·29-s + 0.645·31-s − 1.64·33-s + 1.64·35-s + 9.64·37-s − 2.64·39-s + 4.29·41-s − 1.64·43-s + 2·45-s + 3·47-s − 4.29·49-s − 7.29·51-s − 5.29·53-s + 1.64·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.622·7-s − 0.666·9-s − 0.496·11-s − 0.733·13-s − 0.258·15-s − 1.76·17-s − 1.29·19-s − 0.359·21-s + 0.208·23-s + 0.200·25-s − 0.962·27-s + 0.557·29-s + 0.115·31-s − 0.286·33-s + 0.278·35-s + 1.58·37-s − 0.423·39-s + 0.670·41-s − 0.250·43-s + 0.298·45-s + 0.437·47-s − 0.613·49-s − 1.02·51-s − 0.726·53-s + 0.221·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8762927262\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8762927262\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 + 1.64T + 7T^{2} \) |
| 11 | \( 1 + 1.64T + 11T^{2} \) |
| 13 | \( 1 + 2.64T + 13T^{2} \) |
| 17 | \( 1 + 7.29T + 17T^{2} \) |
| 19 | \( 1 + 5.64T + 19T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 0.645T + 31T^{2} \) |
| 37 | \( 1 - 9.64T + 37T^{2} \) |
| 41 | \( 1 - 4.29T + 41T^{2} \) |
| 43 | \( 1 + 1.64T + 43T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 + 5.29T + 53T^{2} \) |
| 59 | \( 1 + 5.29T + 59T^{2} \) |
| 61 | \( 1 - 8.35T + 61T^{2} \) |
| 67 | \( 1 - 6.93T + 67T^{2} \) |
| 71 | \( 1 - 1.35T + 71T^{2} \) |
| 73 | \( 1 + 4.64T + 73T^{2} \) |
| 79 | \( 1 - 1.64T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 + 1.64T + 89T^{2} \) |
| 97 | \( 1 - 2.93T + 97T^{2} \) |
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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
−8.054866855554606297849069495863, −7.25761466847876599140018357089, −6.50509056478020244294582560548, −6.00674917712227301266124378264, −4.86487080827225321194029574494, −4.35981605019377907361650045009, −3.45852364120179009544075224839, −2.58619303707378642593107364519, −2.21172211855283273085616837390, −0.42095376940755160805644709412,
0.42095376940755160805644709412, 2.21172211855283273085616837390, 2.58619303707378642593107364519, 3.45852364120179009544075224839, 4.35981605019377907361650045009, 4.86487080827225321194029574494, 6.00674917712227301266124378264, 6.50509056478020244294582560548, 7.25761466847876599140018357089, 8.054866855554606297849069495863