L(s) = 1 | + 2.18·3-s + 4.04·5-s − 7-s + 1.76·9-s − 3.98·11-s + 6.74·13-s + 8.82·15-s − 4.92·17-s − 1.24·19-s − 2.18·21-s − 23-s + 11.3·25-s − 2.70·27-s + 6.48·29-s − 8.38·31-s − 8.69·33-s − 4.04·35-s − 7.61·37-s + 14.7·39-s − 7.11·41-s − 1.11·43-s + 7.12·45-s + 8.31·47-s + 49-s − 10.7·51-s − 4.77·53-s − 16.1·55-s + ⋯ |
L(s) = 1 | + 1.25·3-s + 1.80·5-s − 0.377·7-s + 0.587·9-s − 1.20·11-s + 1.87·13-s + 2.27·15-s − 1.19·17-s − 0.286·19-s − 0.476·21-s − 0.208·23-s + 2.26·25-s − 0.519·27-s + 1.20·29-s − 1.50·31-s − 1.51·33-s − 0.683·35-s − 1.25·37-s + 2.35·39-s − 1.11·41-s − 0.170·43-s + 1.06·45-s + 1.21·47-s + 0.142·49-s − 1.50·51-s − 0.655·53-s − 2.17·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.727161199\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.727161199\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 - 2.18T + 3T^{2} \) |
| 5 | \( 1 - 4.04T + 5T^{2} \) |
| 11 | \( 1 + 3.98T + 11T^{2} \) |
| 13 | \( 1 - 6.74T + 13T^{2} \) |
| 17 | \( 1 + 4.92T + 17T^{2} \) |
| 19 | \( 1 + 1.24T + 19T^{2} \) |
| 29 | \( 1 - 6.48T + 29T^{2} \) |
| 31 | \( 1 + 8.38T + 31T^{2} \) |
| 37 | \( 1 + 7.61T + 37T^{2} \) |
| 41 | \( 1 + 7.11T + 41T^{2} \) |
| 43 | \( 1 + 1.11T + 43T^{2} \) |
| 47 | \( 1 - 8.31T + 47T^{2} \) |
| 53 | \( 1 + 4.77T + 53T^{2} \) |
| 59 | \( 1 - 1.31T + 59T^{2} \) |
| 61 | \( 1 - 0.838T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 - 6.46T + 73T^{2} \) |
| 79 | \( 1 + 6.45T + 79T^{2} \) |
| 83 | \( 1 + 5.59T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−10.42029542754967443033398894814, −9.574961492571575205232757838248, −8.740698215727838775750107892871, −8.405488233453608066699893497362, −6.92202312760197568127741131781, −6.09288142992935337203432906932, −5.21424969637098265872416757539, −3.65349872272827095655780785260, −2.60411991707496156001830270983, −1.79632357507754833535704374709,
1.79632357507754833535704374709, 2.60411991707496156001830270983, 3.65349872272827095655780785260, 5.21424969637098265872416757539, 6.09288142992935337203432906932, 6.92202312760197568127741131781, 8.405488233453608066699893497362, 8.740698215727838775750107892871, 9.574961492571575205232757838248, 10.42029542754967443033398894814