| L(s) = 1 | − 0.472·2-s − 1.77·4-s + 3.20·5-s + 4.06·7-s + 1.78·8-s − 1.51·10-s + 11-s − 6.29·13-s − 1.91·14-s + 2.70·16-s − 2.28·17-s + 4.41·19-s − 5.70·20-s − 0.472·22-s − 4.18·23-s + 5.30·25-s + 2.97·26-s − 7.21·28-s − 2.32·29-s + 4.52·31-s − 4.85·32-s + 1.08·34-s + 13.0·35-s − 8.64·37-s − 2.08·38-s + 5.72·40-s − 4.05·41-s + ⋯ |
| L(s) = 1 | − 0.334·2-s − 0.888·4-s + 1.43·5-s + 1.53·7-s + 0.631·8-s − 0.479·10-s + 0.301·11-s − 1.74·13-s − 0.513·14-s + 0.677·16-s − 0.554·17-s + 1.01·19-s − 1.27·20-s − 0.100·22-s − 0.873·23-s + 1.06·25-s + 0.583·26-s − 1.36·28-s − 0.431·29-s + 0.812·31-s − 0.857·32-s + 0.185·34-s + 2.20·35-s − 1.42·37-s − 0.338·38-s + 0.905·40-s − 0.633·41-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
| \(L(1)\) |
\(\approx\) |
\(2.086728465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.086728465\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.472T + 2T^{2} \) |
| 5 | \( 1 - 3.20T + 5T^{2} \) |
| 7 | \( 1 - 4.06T + 7T^{2} \) |
| 13 | \( 1 + 6.29T + 13T^{2} \) |
| 17 | \( 1 + 2.28T + 17T^{2} \) |
| 19 | \( 1 - 4.41T + 19T^{2} \) |
| 23 | \( 1 + 4.18T + 23T^{2} \) |
| 29 | \( 1 + 2.32T + 29T^{2} \) |
| 31 | \( 1 - 4.52T + 31T^{2} \) |
| 37 | \( 1 + 8.64T + 37T^{2} \) |
| 41 | \( 1 + 4.05T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 - 0.746T + 47T^{2} \) |
| 53 | \( 1 - 9.63T + 53T^{2} \) |
| 59 | \( 1 + 2.52T + 59T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 5.36T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 - 1.73T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 - 0.00104T + 97T^{2} \) |
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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
−8.075226891610406312071451682540, −7.56433703519491845141901543527, −6.74997077008871129334748573641, −5.66333300172074242877079310044, −5.13276842132299935766567931031, −4.75504637226340514819664812756, −3.79793018470113761294953996451, −2.34993323057053319642137601868, −1.88906641456530209148834789569, −0.831602916489079541776207650405,
0.831602916489079541776207650405, 1.88906641456530209148834789569, 2.34993323057053319642137601868, 3.79793018470113761294953996451, 4.75504637226340514819664812756, 5.13276842132299935766567931031, 5.66333300172074242877079310044, 6.74997077008871129334748573641, 7.56433703519491845141901543527, 8.075226891610406312071451682540
