| L(s) = 1 | − 2.43·2-s + 3·3-s − 2.05·4-s − 9.35·5-s − 7.31·6-s + 19.3·7-s + 24.5·8-s + 9·9-s + 22.8·10-s − 11·11-s − 6.17·12-s + 25.2·13-s − 47.0·14-s − 28.0·15-s − 43.3·16-s + 24.5·17-s − 21.9·18-s − 79.9·19-s + 19.2·20-s + 57.9·21-s + 26.8·22-s − 10.4·23-s + 73.5·24-s − 37.4·25-s − 61.4·26-s + 27·27-s − 39.7·28-s + ⋯ |
| L(s) = 1 | − 0.861·2-s + 0.577·3-s − 0.257·4-s − 0.836·5-s − 0.497·6-s + 1.04·7-s + 1.08·8-s + 0.333·9-s + 0.721·10-s − 0.301·11-s − 0.148·12-s + 0.537·13-s − 0.898·14-s − 0.483·15-s − 0.676·16-s + 0.349·17-s − 0.287·18-s − 0.965·19-s + 0.215·20-s + 0.601·21-s + 0.259·22-s − 0.0949·23-s + 0.625·24-s − 0.299·25-s − 0.463·26-s + 0.192·27-s − 0.268·28-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 | \( 1 + 2.43T + 8T^{2} \) |
| 5 | \( 1 + 9.35T + 125T^{2} \) |
| 7 | \( 1 - 19.3T + 343T^{2} \) |
| 13 | \( 1 - 25.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 24.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 79.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 10.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 30.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 63.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 25.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 236.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 69.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 323.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 161.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 794.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 486.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 850.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 169.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 481.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.33e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.32e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.65e3T + 9.12e5T^{2} \) |
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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
−8.410055895898857936527635102729, −7.84829682263903171577239325459, −7.39191093555541586535165180800, −6.15536667959398677610737956814, −4.87489754879187022988347140280, −4.33208838482546271353475522292, −3.44170708162154711154325282193, −2.08459824649257264665107448924, −1.16274901342951794241197508569, 0,
1.16274901342951794241197508569, 2.08459824649257264665107448924, 3.44170708162154711154325282193, 4.33208838482546271353475522292, 4.87489754879187022988347140280, 6.15536667959398677610737956814, 7.39191093555541586535165180800, 7.84829682263903171577239325459, 8.410055895898857936527635102729
