| L(s) = 1 | − 1.30·2-s − 0.301·4-s − 3.35·5-s − 0.705·7-s + 2.99·8-s + 4.36·10-s + 0.652·11-s − 6.43·13-s + 0.918·14-s − 3.30·16-s − 2.51·17-s − 3.54·19-s + 1.01·20-s − 0.850·22-s + 0.127·23-s + 6.23·25-s + 8.38·26-s + 0.212·28-s − 3.29·29-s + 0.205·31-s − 1.68·32-s + 3.28·34-s + 2.36·35-s − 11.2·37-s + 4.62·38-s − 10.0·40-s + 0.626·41-s + ⋯ |
| L(s) = 1 | − 0.921·2-s − 0.150·4-s − 1.49·5-s − 0.266·7-s + 1.06·8-s + 1.38·10-s + 0.196·11-s − 1.78·13-s + 0.245·14-s − 0.826·16-s − 0.610·17-s − 0.814·19-s + 0.225·20-s − 0.181·22-s + 0.0265·23-s + 1.24·25-s + 1.64·26-s + 0.0401·28-s − 0.611·29-s + 0.0369·31-s − 0.298·32-s + 0.563·34-s + 0.399·35-s − 1.85·37-s + 0.750·38-s − 1.58·40-s + 0.0979·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06056411619\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06056411619\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 2 | \( 1 + 1.30T + 2T^{2} \) |
| 5 | \( 1 + 3.35T + 5T^{2} \) |
| 7 | \( 1 + 0.705T + 7T^{2} \) |
| 11 | \( 1 - 0.652T + 11T^{2} \) |
| 13 | \( 1 + 6.43T + 13T^{2} \) |
| 17 | \( 1 + 2.51T + 17T^{2} \) |
| 19 | \( 1 + 3.54T + 19T^{2} \) |
| 23 | \( 1 - 0.127T + 23T^{2} \) |
| 29 | \( 1 + 3.29T + 29T^{2} \) |
| 31 | \( 1 - 0.205T + 31T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 - 0.626T + 41T^{2} \) |
| 43 | \( 1 + 1.83T + 43T^{2} \) |
| 47 | \( 1 - 2.95T + 47T^{2} \) |
| 53 | \( 1 - 6.27T + 53T^{2} \) |
| 59 | \( 1 + 5.90T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 + 6.73T + 71T^{2} \) |
| 73 | \( 1 + 3.36T + 73T^{2} \) |
| 79 | \( 1 - 6.77T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 + 13.7T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−8.449447592360840048460183373873, −7.76541167770344701988852503429, −7.25602883082615019628292006164, −6.67322031477869753512565864650, −5.27498420949837926889956891026, −4.51168184435931496902564207974, −4.00279364141728096097270948289, −2.92554172172916855495397589527, −1.74370163479746484229103258127, −0.15806523089641708749839499913,
0.15806523089641708749839499913, 1.74370163479746484229103258127, 2.92554172172916855495397589527, 4.00279364141728096097270948289, 4.51168184435931496902564207974, 5.27498420949837926889956891026, 6.67322031477869753512565864650, 7.25602883082615019628292006164, 7.76541167770344701988852503429, 8.449447592360840048460183373873
