| L(s) = 1 | − 0.784·2-s − 1.38·4-s − 2.15·5-s − 2.00·7-s + 2.65·8-s + 1.69·10-s − 0.548·11-s + 0.815·13-s + 1.57·14-s + 0.684·16-s − 4.17·17-s − 0.0461·19-s + 2.98·20-s + 0.430·22-s + 23-s − 0.357·25-s − 0.639·26-s + 2.77·28-s − 29-s + 0.466·31-s − 5.84·32-s + 3.27·34-s + 4.32·35-s − 10.1·37-s + 0.0362·38-s − 5.72·40-s − 2.16·41-s + ⋯ |
| L(s) = 1 | − 0.554·2-s − 0.692·4-s − 0.963·5-s − 0.758·7-s + 0.938·8-s + 0.534·10-s − 0.165·11-s + 0.226·13-s + 0.421·14-s + 0.171·16-s − 1.01·17-s − 0.0105·19-s + 0.666·20-s + 0.0918·22-s + 0.208·23-s − 0.0715·25-s − 0.125·26-s + 0.525·28-s − 0.185·29-s + 0.0838·31-s − 1.03·32-s + 0.562·34-s + 0.731·35-s − 1.67·37-s + 0.00588·38-s − 0.904·40-s − 0.338·41-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
| \(L(1)\) |
\(\approx\) |
\(0.2971284248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2971284248\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.784T + 2T^{2} \) |
| 5 | \( 1 + 2.15T + 5T^{2} \) |
| 7 | \( 1 + 2.00T + 7T^{2} \) |
| 11 | \( 1 + 0.548T + 11T^{2} \) |
| 13 | \( 1 - 0.815T + 13T^{2} \) |
| 17 | \( 1 + 4.17T + 17T^{2} \) |
| 19 | \( 1 + 0.0461T + 19T^{2} \) |
| 31 | \( 1 - 0.466T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 + 2.16T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 0.0241T + 47T^{2} \) |
| 53 | \( 1 + 3.55T + 53T^{2} \) |
| 59 | \( 1 + 9.41T + 59T^{2} \) |
| 61 | \( 1 + 7.51T + 61T^{2} \) |
| 67 | \( 1 + 3.60T + 67T^{2} \) |
| 71 | \( 1 - 9.01T + 71T^{2} \) |
| 73 | \( 1 - 6.72T + 73T^{2} \) |
| 79 | \( 1 + 5.39T + 79T^{2} \) |
| 83 | \( 1 + 7.98T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 + 1.00T + 97T^{2} \) |
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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
−8.202269384301950671457016147275, −7.47217463283465583754469417490, −6.89391081106745339926463377069, −6.01720952629479538795430795373, −5.09043882332070060379371227466, −4.31462593561076013860611848264, −3.75656040801687208788926234166, −2.90738396851161569884428604437, −1.61299551420511433086816109751, −0.31565424198136708231180546276,
0.31565424198136708231180546276, 1.61299551420511433086816109751, 2.90738396851161569884428604437, 3.75656040801687208788926234166, 4.31462593561076013860611848264, 5.09043882332070060379371227466, 6.01720952629479538795430795373, 6.89391081106745339926463377069, 7.47217463283465583754469417490, 8.202269384301950671457016147275
