| L(s) = 1 | − 2-s + 4-s + 3.80·5-s + 0.0489·7-s − 8-s − 3.80·10-s + 4.82·11-s + 2.29·13-s − 0.0489·14-s + 16-s + 7.51·17-s − 1.46·19-s + 3.80·20-s − 4.82·22-s + 0.951·23-s + 9.45·25-s − 2.29·26-s + 0.0489·28-s + 8.54·29-s − 3.24·31-s − 32-s − 7.51·34-s + 0.185·35-s + 1.26·37-s + 1.46·38-s − 3.80·40-s + 2.91·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.70·5-s + 0.0184·7-s − 0.353·8-s − 1.20·10-s + 1.45·11-s + 0.636·13-s − 0.0130·14-s + 0.250·16-s + 1.82·17-s − 0.336·19-s + 0.850·20-s − 1.02·22-s + 0.198·23-s + 1.89·25-s − 0.450·26-s + 0.00924·28-s + 1.58·29-s − 0.583·31-s − 0.176·32-s − 1.28·34-s + 0.0314·35-s + 0.208·37-s + 0.237·38-s − 0.601·40-s + 0.455·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.484845965\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.484845965\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 223 | \( 1 + T \) |
| good | 5 | \( 1 - 3.80T + 5T^{2} \) |
| 7 | \( 1 - 0.0489T + 7T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 - 2.29T + 13T^{2} \) |
| 17 | \( 1 - 7.51T + 17T^{2} \) |
| 19 | \( 1 + 1.46T + 19T^{2} \) |
| 23 | \( 1 - 0.951T + 23T^{2} \) |
| 29 | \( 1 - 8.54T + 29T^{2} \) |
| 31 | \( 1 + 3.24T + 31T^{2} \) |
| 37 | \( 1 - 1.26T + 37T^{2} \) |
| 41 | \( 1 - 2.91T + 41T^{2} \) |
| 43 | \( 1 + 3.35T + 43T^{2} \) |
| 47 | \( 1 + 9.67T + 47T^{2} \) |
| 53 | \( 1 + 8.94T + 53T^{2} \) |
| 59 | \( 1 + 1.86T + 59T^{2} \) |
| 61 | \( 1 + 2.62T + 61T^{2} \) |
| 67 | \( 1 - 6.47T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 + 2.18T + 73T^{2} \) |
| 79 | \( 1 + 17.2T + 79T^{2} \) |
| 83 | \( 1 - 4.85T + 83T^{2} \) |
| 89 | \( 1 - 8.66T + 89T^{2} \) |
| 97 | \( 1 - 9.12T + 97T^{2} \) |
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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
−8.553641181026917244839313572359, −7.905803610351472958456000002765, −6.69766172857113772107957947816, −6.41697491982460175911877950557, −5.71013920077517508520672858432, −4.88860903308283196774950605932, −3.61639473919928690485618783147, −2.78393909818623137328407463506, −1.56948868269842477345183718666, −1.20516524741412933082134815532,
1.20516524741412933082134815532, 1.56948868269842477345183718666, 2.78393909818623137328407463506, 3.61639473919928690485618783147, 4.88860903308283196774950605932, 5.71013920077517508520672858432, 6.41697491982460175911877950557, 6.69766172857113772107957947816, 7.905803610351472958456000002765, 8.553641181026917244839313572359
