| L(s) = 1 | + 1.91·3-s + 3.07·5-s − 0.591·7-s + 0.673·9-s + 1.99·11-s + 3.60·13-s + 5.89·15-s + 2.69·17-s − 1.18·19-s − 1.13·21-s − 8.54·23-s + 4.44·25-s − 4.45·27-s + 6.20·29-s − 7.60·31-s + 3.81·33-s − 1.81·35-s + 10.3·37-s + 6.91·39-s + 8.86·41-s + 6.29·43-s + 2.07·45-s + 1.22·47-s − 6.64·49-s + 5.16·51-s + 8.63·53-s + 6.11·55-s + ⋯ |
| L(s) = 1 | + 1.10·3-s + 1.37·5-s − 0.223·7-s + 0.224·9-s + 0.600·11-s + 1.00·13-s + 1.52·15-s + 0.653·17-s − 0.272·19-s − 0.247·21-s − 1.78·23-s + 0.888·25-s − 0.858·27-s + 1.15·29-s − 1.36·31-s + 0.664·33-s − 0.307·35-s + 1.69·37-s + 1.10·39-s + 1.38·41-s + 0.960·43-s + 0.308·45-s + 0.179·47-s − 0.949·49-s + 0.722·51-s + 1.18·53-s + 0.824·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.232921896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.232921896\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 751 | \( 1 + T \) |
| good | 3 | \( 1 - 1.91T + 3T^{2} \) |
| 5 | \( 1 - 3.07T + 5T^{2} \) |
| 7 | \( 1 + 0.591T + 7T^{2} \) |
| 11 | \( 1 - 1.99T + 11T^{2} \) |
| 13 | \( 1 - 3.60T + 13T^{2} \) |
| 17 | \( 1 - 2.69T + 17T^{2} \) |
| 19 | \( 1 + 1.18T + 19T^{2} \) |
| 23 | \( 1 + 8.54T + 23T^{2} \) |
| 29 | \( 1 - 6.20T + 29T^{2} \) |
| 31 | \( 1 + 7.60T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 - 8.86T + 41T^{2} \) |
| 43 | \( 1 - 6.29T + 43T^{2} \) |
| 47 | \( 1 - 1.22T + 47T^{2} \) |
| 53 | \( 1 - 8.63T + 53T^{2} \) |
| 59 | \( 1 - 3.53T + 59T^{2} \) |
| 61 | \( 1 + 7.16T + 61T^{2} \) |
| 67 | \( 1 + 0.813T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 - 0.290T + 83T^{2} \) |
| 89 | \( 1 + 9.51T + 89T^{2} \) |
| 97 | \( 1 - 8.90T + 97T^{2} \) |
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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
−8.148427695085820608594193439681, −7.59292005591894245899367424534, −6.41210966692948824228608073176, −6.08030655339228525349768718554, −5.42874632408269754735825810936, −4.14545050692402616964158416612, −3.62666125855181574866744977451, −2.59893376445593150377833636543, −2.06617535905145384842559359527, −1.09036341323575997288384514950,
1.09036341323575997288384514950, 2.06617535905145384842559359527, 2.59893376445593150377833636543, 3.62666125855181574866744977451, 4.14545050692402616964158416612, 5.42874632408269754735825810936, 6.08030655339228525349768718554, 6.41210966692948824228608073176, 7.59292005591894245899367424534, 8.148427695085820608594193439681
