| L(s) = 1 | + 2-s + 3-s + 4-s + 2.08·5-s + 6-s + 1.83·7-s + 8-s + 9-s + 2.08·10-s − 1.01·11-s + 12-s + 5.87·13-s + 1.83·14-s + 2.08·15-s + 16-s + 5.63·17-s + 18-s + 19-s + 2.08·20-s + 1.83·21-s − 1.01·22-s + 3.02·23-s + 24-s − 0.650·25-s + 5.87·26-s + 27-s + 1.83·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.932·5-s + 0.408·6-s + 0.693·7-s + 0.353·8-s + 0.333·9-s + 0.659·10-s − 0.304·11-s + 0.288·12-s + 1.62·13-s + 0.490·14-s + 0.538·15-s + 0.250·16-s + 1.36·17-s + 0.235·18-s + 0.229·19-s + 0.466·20-s + 0.400·21-s − 0.215·22-s + 0.630·23-s + 0.204·24-s − 0.130·25-s + 1.15·26-s + 0.192·27-s + 0.346·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.005757221\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.005757221\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| good | 5 | \( 1 - 2.08T + 5T^{2} \) |
| 7 | \( 1 - 1.83T + 7T^{2} \) |
| 11 | \( 1 + 1.01T + 11T^{2} \) |
| 13 | \( 1 - 5.87T + 13T^{2} \) |
| 17 | \( 1 - 5.63T + 17T^{2} \) |
| 23 | \( 1 - 3.02T + 23T^{2} \) |
| 29 | \( 1 - 5.04T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 6.50T + 37T^{2} \) |
| 41 | \( 1 + 2.77T + 41T^{2} \) |
| 43 | \( 1 - 2.85T + 43T^{2} \) |
| 47 | \( 1 + 4.87T + 47T^{2} \) |
| 59 | \( 1 + 5.48T + 59T^{2} \) |
| 61 | \( 1 + 0.773T + 61T^{2} \) |
| 67 | \( 1 - 1.08T + 67T^{2} \) |
| 71 | \( 1 + 7.41T + 71T^{2} \) |
| 73 | \( 1 + 6.01T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + 17.2T + 83T^{2} \) |
| 89 | \( 1 - 18.7T + 89T^{2} \) |
| 97 | \( 1 + 2.69T + 97T^{2} \) |
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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
−8.045416899819269463184175067460, −7.37199243842365797085770986721, −6.54696158643940458719431136655, −5.71188090481463620553931791022, −5.37232482232439626180944992412, −4.43607565639447157743884553718, −3.49358021022766142236289857336, −2.99720526055517272012703916468, −1.77401049007537515370807728196, −1.35364538412557692515924861690,
1.35364538412557692515924861690, 1.77401049007537515370807728196, 2.99720526055517272012703916468, 3.49358021022766142236289857336, 4.43607565639447157743884553718, 5.37232482232439626180944992412, 5.71188090481463620553931791022, 6.54696158643940458719431136655, 7.37199243842365797085770986721, 8.045416899819269463184175067460
