| L(s) = 1 | − 1.51·2-s + 0.295·4-s + 1.86·5-s + 1.57·7-s + 2.58·8-s − 2.82·10-s − 0.419·11-s + 4.28·13-s − 2.38·14-s − 4.50·16-s − 1.70·17-s − 5.83·19-s + 0.551·20-s + 0.636·22-s − 23-s − 1.51·25-s − 6.49·26-s + 0.464·28-s − 29-s − 2.45·31-s + 1.65·32-s + 2.58·34-s + 2.93·35-s − 3.07·37-s + 8.84·38-s + 4.82·40-s + 9.46·41-s + ⋯ |
| L(s) = 1 | − 1.07·2-s + 0.147·4-s + 0.834·5-s + 0.593·7-s + 0.912·8-s − 0.894·10-s − 0.126·11-s + 1.18·13-s − 0.636·14-s − 1.12·16-s − 0.413·17-s − 1.33·19-s + 0.123·20-s + 0.135·22-s − 0.208·23-s − 0.303·25-s − 1.27·26-s + 0.0877·28-s − 0.185·29-s − 0.441·31-s + 0.293·32-s + 0.442·34-s + 0.495·35-s − 0.505·37-s + 1.43·38-s + 0.762·40-s + 1.47·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.249084780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.249084780\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 1.51T + 2T^{2} \) |
| 5 | \( 1 - 1.86T + 5T^{2} \) |
| 7 | \( 1 - 1.57T + 7T^{2} \) |
| 11 | \( 1 + 0.419T + 11T^{2} \) |
| 13 | \( 1 - 4.28T + 13T^{2} \) |
| 17 | \( 1 + 1.70T + 17T^{2} \) |
| 19 | \( 1 + 5.83T + 19T^{2} \) |
| 31 | \( 1 + 2.45T + 31T^{2} \) |
| 37 | \( 1 + 3.07T + 37T^{2} \) |
| 41 | \( 1 - 9.46T + 41T^{2} \) |
| 43 | \( 1 + 3.20T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 - 5.61T + 53T^{2} \) |
| 59 | \( 1 + 2.16T + 59T^{2} \) |
| 61 | \( 1 - 5.23T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 9.31T + 71T^{2} \) |
| 73 | \( 1 - 15.9T + 73T^{2} \) |
| 79 | \( 1 - 5.67T + 79T^{2} \) |
| 83 | \( 1 - 3.89T + 83T^{2} \) |
| 89 | \( 1 - 6.42T + 89T^{2} \) |
| 97 | \( 1 - 2.73T + 97T^{2} \) |
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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
−8.186229313548228816491788514965, −7.66591353833828023986985693036, −6.71459201347528533860331996569, −6.07530430249453879505496469979, −5.31004855472913535424914050302, −4.41592624948539382872632009721, −3.74462250829620736519922825864, −2.25452938494520594431259026717, −1.77806326085170616051252296932, −0.71157951596913940535758396694,
0.71157951596913940535758396694, 1.77806326085170616051252296932, 2.25452938494520594431259026717, 3.74462250829620736519922825864, 4.41592624948539382872632009721, 5.31004855472913535424914050302, 6.07530430249453879505496469979, 6.71459201347528533860331996569, 7.66591353833828023986985693036, 8.186229313548228816491788514965
