| L(s) = 1 | − 5.57·2-s − 3·3-s + 23.0·4-s − 16.5·5-s + 16.7·6-s − 30.4·7-s − 83.8·8-s + 9·9-s + 92.1·10-s + 11·11-s − 69.1·12-s − 61.2·13-s + 169.·14-s + 49.5·15-s + 282.·16-s − 7.29·17-s − 50.1·18-s − 92.9·19-s − 381.·20-s + 91.2·21-s − 61.2·22-s + 17.5·23-s + 251.·24-s + 148.·25-s + 341.·26-s − 27·27-s − 701.·28-s + ⋯ |
| L(s) = 1 | − 1.97·2-s − 0.577·3-s + 2.88·4-s − 1.47·5-s + 1.13·6-s − 1.64·7-s − 3.70·8-s + 0.333·9-s + 2.91·10-s + 0.301·11-s − 1.66·12-s − 1.30·13-s + 3.23·14-s + 0.853·15-s + 4.42·16-s − 0.104·17-s − 0.656·18-s − 1.12·19-s − 4.26·20-s + 0.948·21-s − 0.594·22-s + 0.159·23-s + 2.13·24-s + 1.18·25-s + 2.57·26-s − 0.192·27-s − 4.73·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 + 5.57T + 8T^{2} \) |
| 5 | \( 1 + 16.5T + 125T^{2} \) |
| 7 | \( 1 + 30.4T + 343T^{2} \) |
| 13 | \( 1 + 61.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 7.29T + 4.91e3T^{2} \) |
| 19 | \( 1 + 92.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 17.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 89.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 312.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 383.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 327.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 285.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 367.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 655.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 279.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 215.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 183.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 592.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.09e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.25e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 547.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.49e3T + 9.12e5T^{2} \) |
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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
−8.606734047593008299115903298724, −7.38949461309524043050022116998, −7.26035198133039764705681453534, −6.53821234689047434168835440528, −5.60616879033205786271348697478, −3.97555060646947747469006002373, −3.14948785575482248940097959740, −2.05985439687800662431826333731, −0.49850592410032629031373373851, 0,
0.49850592410032629031373373851, 2.05985439687800662431826333731, 3.14948785575482248940097959740, 3.97555060646947747469006002373, 5.60616879033205786271348697478, 6.53821234689047434168835440528, 7.26035198133039764705681453534, 7.38949461309524043050022116998, 8.606734047593008299115903298724
