| L(s) = 1 | + i·2-s − 4-s + 5-s + (−2.56 + 0.648i)7-s − i·8-s + i·10-s + 1.29i·11-s + 3.13i·13-s + (−0.648 − 2.56i)14-s + 16-s − 5.53·17-s + 7.37i·19-s − 20-s − 1.29·22-s − 1.83i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.447·5-s + (−0.969 + 0.245i)7-s − 0.353i·8-s + 0.316i·10-s + 0.390i·11-s + 0.868i·13-s + (−0.173 − 0.685i)14-s + 0.250·16-s − 1.34·17-s + 1.69i·19-s − 0.223·20-s − 0.276·22-s − 0.382i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.933 - 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.162016 + 0.870731i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.162016 + 0.870731i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (2.56 - 0.648i)T \) |
| good | 11 | \( 1 - 1.29iT - 11T^{2} \) |
| 13 | \( 1 - 3.13iT - 13T^{2} \) |
| 17 | \( 1 + 5.53T + 17T^{2} \) |
| 19 | \( 1 - 7.37iT - 19T^{2} \) |
| 23 | \( 1 + 1.83iT - 23T^{2} \) |
| 29 | \( 1 + 1.83iT - 29T^{2} \) |
| 31 | \( 1 - 10.4iT - 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 + 3.13T + 41T^{2} \) |
| 43 | \( 1 - 3.53T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 + 4.42iT - 53T^{2} \) |
| 59 | \( 1 + 7.18T + 59T^{2} \) |
| 61 | \( 1 + 4.88iT - 61T^{2} \) |
| 67 | \( 1 + 9.79T + 67T^{2} \) |
| 71 | \( 1 - 7.37iT - 71T^{2} \) |
| 73 | \( 1 - 3.40iT - 73T^{2} \) |
| 79 | \( 1 - 9.01T + 79T^{2} \) |
| 83 | \( 1 - 6.26T + 83T^{2} \) |
| 89 | \( 1 - 7.94T + 89T^{2} \) |
| 97 | \( 1 + 8.09iT - 97T^{2} \) |
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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
−10.68967658357673796443891083719, −9.992801426752247027585070214542, −9.090854090903368564047981650120, −8.535104926453170563613688039621, −7.15885678651247408929657663846, −6.56584410357668321079694033870, −5.75431476127640882543693970866, −4.61628197513545905794047586183, −3.50810823013308965636214770699, −1.96447134589460294068535491115,
0.46348315504695558310538728789, 2.33922142995592365519542965104, 3.27262921579458301335360498814, 4.45431590941835014794398187480, 5.60472635234434195215378991601, 6.55623560265222768483337902268, 7.55142564371353429690156807536, 8.969146861451722838539985861460, 9.238038297027291871319447232485, 10.43188970158167922672751476023
