| L(s) = 1 | − 0.423·2-s − 1.82·4-s + 1.36·5-s + 0.865·7-s + 1.61·8-s − 0.577·10-s + 11-s + 2.63·13-s − 0.366·14-s + 2.95·16-s + 3.12·17-s − 0.281·19-s − 2.48·20-s − 0.423·22-s − 1.73·23-s − 3.13·25-s − 1.11·26-s − 1.57·28-s + 7.07·29-s − 4.28·31-s − 4.48·32-s − 1.32·34-s + 1.18·35-s + 3.94·37-s + 0.119·38-s + 2.20·40-s + 7.38·41-s + ⋯ |
| L(s) = 1 | − 0.299·2-s − 0.910·4-s + 0.610·5-s + 0.327·7-s + 0.571·8-s − 0.182·10-s + 0.301·11-s + 0.731·13-s − 0.0978·14-s + 0.739·16-s + 0.758·17-s − 0.0646·19-s − 0.555·20-s − 0.0902·22-s − 0.362·23-s − 0.627·25-s − 0.218·26-s − 0.297·28-s + 1.31·29-s − 0.769·31-s − 0.792·32-s − 0.226·34-s + 0.199·35-s + 0.648·37-s + 0.0193·38-s + 0.348·40-s + 1.15·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.783105789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.783105789\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 0.423T + 2T^{2} \) |
| 5 | \( 1 - 1.36T + 5T^{2} \) |
| 7 | \( 1 - 0.865T + 7T^{2} \) |
| 13 | \( 1 - 2.63T + 13T^{2} \) |
| 17 | \( 1 - 3.12T + 17T^{2} \) |
| 19 | \( 1 + 0.281T + 19T^{2} \) |
| 23 | \( 1 + 1.73T + 23T^{2} \) |
| 29 | \( 1 - 7.07T + 29T^{2} \) |
| 31 | \( 1 + 4.28T + 31T^{2} \) |
| 37 | \( 1 - 3.94T + 37T^{2} \) |
| 41 | \( 1 - 7.38T + 41T^{2} \) |
| 43 | \( 1 - 5.77T + 43T^{2} \) |
| 47 | \( 1 - 3.73T + 47T^{2} \) |
| 53 | \( 1 - 5.86T + 53T^{2} \) |
| 59 | \( 1 - 7.96T + 59T^{2} \) |
| 67 | \( 1 - 4.80T + 67T^{2} \) |
| 71 | \( 1 - 7.35T + 71T^{2} \) |
| 73 | \( 1 + 16.3T + 73T^{2} \) |
| 79 | \( 1 + 6.03T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−8.111958474173094183514397596628, −7.62162371816702820660061658168, −6.60898072637910167652773389065, −5.78261482839664775677425041062, −5.36410079002657924091978570877, −4.31569663467331829046113629422, −3.85813994159362833620319653886, −2.73458441929356996418710970212, −1.60828153744514781242236761140, −0.800569395488316685797184112216,
0.800569395488316685797184112216, 1.60828153744514781242236761140, 2.73458441929356996418710970212, 3.85813994159362833620319653886, 4.31569663467331829046113629422, 5.36410079002657924091978570877, 5.78261482839664775677425041062, 6.60898072637910167652773389065, 7.62162371816702820660061658168, 8.111958474173094183514397596628
