| L(s) = 1 | + 1.73i·7-s − 2.44·11-s + 13-s − 4.24i·17-s − 1.73i·19-s − 2.44·23-s + 4.24i·29-s + 5.19i·31-s − 4·37-s + 5.19i·43-s − 12.2·47-s + 4·49-s + 12.2·59-s − 7·61-s + 8.66i·67-s + ⋯ |
| L(s) = 1 | + 0.654i·7-s − 0.738·11-s + 0.277·13-s − 1.02i·17-s − 0.397i·19-s − 0.510·23-s + 0.787i·29-s + 0.933i·31-s − 0.657·37-s + 0.792i·43-s − 1.78·47-s + 0.571·49-s + 1.59·59-s − 0.896·61-s + 1.05i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2352710185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2352710185\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 1.73iT - 7T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + 4.24iT - 17T^{2} \) |
| 19 | \( 1 + 1.73iT - 19T^{2} \) |
| 23 | \( 1 + 2.44T + 23T^{2} \) |
| 29 | \( 1 - 4.24iT - 29T^{2} \) |
| 31 | \( 1 - 5.19iT - 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 5.19iT - 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 - 8.66iT - 67T^{2} \) |
| 71 | \( 1 + 9.79T + 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 + 6.92iT - 79T^{2} \) |
| 83 | \( 1 + 7.34T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 11T + 97T^{2} \) |
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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
−8.773017224592351588045436831229, −8.333973536272238745048328296242, −7.36295227244939126895240868615, −6.78655197335690337424999985556, −5.79629272134014851436876999381, −5.19645003817239496368203240069, −4.45893872800144472672677492531, −3.23746419778675578238108836616, −2.63166567989812121227587478247, −1.48050069500457330584241053561,
0.06742380069697705155858769149, 1.47910703050578441380962904554, 2.48970827325520582021224754910, 3.65026269860245620464269706201, 4.18093807253486408455357243290, 5.21465385400911639399041264878, 5.96993295699054669015993061163, 6.67620117713076562233170694403, 7.61877295566907264019942403142, 8.081273960741569043142387862195
