| L(s) = 1 | + 0.697·2-s + 2.89·3-s − 1.51·4-s + 0.682·5-s + 2.01·6-s − 2.45·8-s + 5.36·9-s + 0.476·10-s − 5.51·11-s − 4.37·12-s + 2.82·13-s + 1.97·15-s + 1.31·16-s + 4.54·17-s + 3.74·18-s + 0.369·19-s − 1.03·20-s − 3.84·22-s + 8.06·23-s − 7.08·24-s − 4.53·25-s + 1.96·26-s + 6.82·27-s + 5.46·29-s + 1.37·30-s − 6.64·31-s + 5.82·32-s + ⋯ |
| L(s) = 1 | + 0.493·2-s + 1.66·3-s − 0.756·4-s + 0.305·5-s + 0.823·6-s − 0.866·8-s + 1.78·9-s + 0.150·10-s − 1.66·11-s − 1.26·12-s + 0.782·13-s + 0.509·15-s + 0.328·16-s + 1.10·17-s + 0.881·18-s + 0.0848·19-s − 0.230·20-s − 0.820·22-s + 1.68·23-s − 1.44·24-s − 0.906·25-s + 0.386·26-s + 1.31·27-s + 1.01·29-s + 0.251·30-s − 1.19·31-s + 1.02·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.122904389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.122904389\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 0.697T + 2T^{2} \) |
| 3 | \( 1 - 2.89T + 3T^{2} \) |
| 5 | \( 1 - 0.682T + 5T^{2} \) |
| 11 | \( 1 + 5.51T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 - 4.54T + 17T^{2} \) |
| 19 | \( 1 - 0.369T + 19T^{2} \) |
| 23 | \( 1 - 8.06T + 23T^{2} \) |
| 29 | \( 1 - 5.46T + 29T^{2} \) |
| 31 | \( 1 + 6.64T + 31T^{2} \) |
| 37 | \( 1 + 1.54T + 37T^{2} \) |
| 41 | \( 1 + 0.605T + 41T^{2} \) |
| 43 | \( 1 - 5.51T + 43T^{2} \) |
| 47 | \( 1 + 0.714T + 47T^{2} \) |
| 53 | \( 1 - 8.91T + 53T^{2} \) |
| 59 | \( 1 + 0.745T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 + 0.584T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 - 8.50T + 73T^{2} \) |
| 79 | \( 1 - 6.70T + 79T^{2} \) |
| 83 | \( 1 + 5.60T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 - 1.29T + 97T^{2} \) |
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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
−8.107798504550355212781294322738, −7.64745812061874263523519337101, −6.76972050393263385262203687718, −5.51032953791668634929041265419, −5.28815997102633435716683073253, −4.20559700195944140085098070050, −3.49095616371594348224470609863, −2.96163983321091180503679617999, −2.21213571966707416949134983719, −0.930745378510759715914557812985,
0.930745378510759715914557812985, 2.21213571966707416949134983719, 2.96163983321091180503679617999, 3.49095616371594348224470609863, 4.20559700195944140085098070050, 5.28815997102633435716683073253, 5.51032953791668634929041265419, 6.76972050393263385262203687718, 7.64745812061874263523519337101, 8.107798504550355212781294322738
