| L(s) = 1 | − 2-s + 4-s + 0.740·5-s + 4.13·7-s − 8-s − 0.740·10-s − 4.54·11-s + 0.865·13-s − 4.13·14-s + 16-s + 4.35·17-s + 1.55·19-s + 0.740·20-s + 4.54·22-s + 3.50·23-s − 4.45·25-s − 0.865·26-s + 4.13·28-s + 3.26·29-s + 10.6·31-s − 32-s − 4.35·34-s + 3.06·35-s − 1.76·37-s − 1.55·38-s − 0.740·40-s + 2.57·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.331·5-s + 1.56·7-s − 0.353·8-s − 0.234·10-s − 1.37·11-s + 0.239·13-s − 1.10·14-s + 0.250·16-s + 1.05·17-s + 0.356·19-s + 0.165·20-s + 0.969·22-s + 0.730·23-s − 0.890·25-s − 0.169·26-s + 0.781·28-s + 0.606·29-s + 1.92·31-s − 0.176·32-s − 0.747·34-s + 0.517·35-s − 0.289·37-s − 0.252·38-s − 0.117·40-s + 0.402·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.534116037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.534116037\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 0.740T + 5T^{2} \) |
| 7 | \( 1 - 4.13T + 7T^{2} \) |
| 11 | \( 1 + 4.54T + 11T^{2} \) |
| 13 | \( 1 - 0.865T + 13T^{2} \) |
| 17 | \( 1 - 4.35T + 17T^{2} \) |
| 19 | \( 1 - 1.55T + 19T^{2} \) |
| 23 | \( 1 - 3.50T + 23T^{2} \) |
| 29 | \( 1 - 3.26T + 29T^{2} \) |
| 31 | \( 1 - 10.6T + 31T^{2} \) |
| 37 | \( 1 + 1.76T + 37T^{2} \) |
| 41 | \( 1 - 2.57T + 41T^{2} \) |
| 43 | \( 1 + 2.75T + 43T^{2} \) |
| 47 | \( 1 + 9.69T + 47T^{2} \) |
| 53 | \( 1 + 4.00T + 53T^{2} \) |
| 59 | \( 1 + 1.43T + 59T^{2} \) |
| 61 | \( 1 + 4.32T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 - 5.08T + 71T^{2} \) |
| 73 | \( 1 + 0.573T + 73T^{2} \) |
| 79 | \( 1 - 6.75T + 79T^{2} \) |
| 83 | \( 1 + 9.22T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 - 5.75T + 97T^{2} \) |
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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
−9.648977994720856326618710072516, −8.426070525782463380807337695824, −8.065363659020796089557678360827, −7.45218135692178299811824137947, −6.27572572404915945113641459856, −5.29834555179229101902497540978, −4.73989204286020010404777768719, −3.17730595041556978709258516176, −2.12923294530706400881745642473, −1.04918511319883005299758043257,
1.04918511319883005299758043257, 2.12923294530706400881745642473, 3.17730595041556978709258516176, 4.73989204286020010404777768719, 5.29834555179229101902497540978, 6.27572572404915945113641459856, 7.45218135692178299811824137947, 8.065363659020796089557678360827, 8.426070525782463380807337695824, 9.648977994720856326618710072516
