| L(s) = 1 | + 2-s + 4-s − 2.37·5-s − 7-s + 8-s − 2.37·10-s + 2.37·11-s − 13-s − 14-s + 16-s + 4.37·17-s + 1.62·19-s − 2.37·20-s + 2.37·22-s + 3.62·23-s + 0.627·25-s − 26-s − 28-s + 6.37·29-s − 4.74·31-s + 32-s + 4.37·34-s + 2.37·35-s − 4.37·37-s + 1.62·38-s − 2.37·40-s + 8.74·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.06·5-s − 0.377·7-s + 0.353·8-s − 0.750·10-s + 0.715·11-s − 0.277·13-s − 0.267·14-s + 0.250·16-s + 1.06·17-s + 0.373·19-s − 0.530·20-s + 0.505·22-s + 0.756·23-s + 0.125·25-s − 0.196·26-s − 0.188·28-s + 1.18·29-s − 0.852·31-s + 0.176·32-s + 0.749·34-s + 0.400·35-s − 0.718·37-s + 0.264·38-s − 0.375·40-s + 1.36·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.260385246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.260385246\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2.37T + 5T^{2} \) |
| 11 | \( 1 - 2.37T + 11T^{2} \) |
| 17 | \( 1 - 4.37T + 17T^{2} \) |
| 19 | \( 1 - 1.62T + 19T^{2} \) |
| 23 | \( 1 - 3.62T + 23T^{2} \) |
| 29 | \( 1 - 6.37T + 29T^{2} \) |
| 31 | \( 1 + 4.74T + 31T^{2} \) |
| 37 | \( 1 + 4.37T + 37T^{2} \) |
| 41 | \( 1 - 8.74T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 1.25T + 47T^{2} \) |
| 53 | \( 1 - 8.74T + 53T^{2} \) |
| 59 | \( 1 - 2T + 59T^{2} \) |
| 61 | \( 1 - 5.11T + 61T^{2} \) |
| 67 | \( 1 - 9.48T + 67T^{2} \) |
| 71 | \( 1 + 4.74T + 71T^{2} \) |
| 73 | \( 1 + 8.37T + 73T^{2} \) |
| 79 | \( 1 - 4.74T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 3.25T + 89T^{2} \) |
| 97 | \( 1 - 7.48T + 97T^{2} \) |
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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
−9.373857463487479114668721141890, −8.511085555264429188428361004414, −7.51167601164705766351562169422, −7.10947571004955242301607573170, −6.06076001080333560473021100537, −5.22538541035983250319292524494, −4.18261892703617307841785715116, −3.59714856281968611883874867388, −2.64434025492289001675448653364, −0.979531254790084051525936085842,
0.979531254790084051525936085842, 2.64434025492289001675448653364, 3.59714856281968611883874867388, 4.18261892703617307841785715116, 5.22538541035983250319292524494, 6.06076001080333560473021100537, 7.10947571004955242301607573170, 7.51167601164705766351562169422, 8.511085555264429188428361004414, 9.373857463487479114668721141890
