| L(s) = 1 | + 3.73·2-s − 3·3-s + 5.95·4-s − 11.2·6-s − 36.4·7-s − 7.64·8-s + 9·9-s + 19.1·11-s − 17.8·12-s − 13·13-s − 136.·14-s − 76.1·16-s + 83.8·17-s + 33.6·18-s + 46.8·19-s + 109.·21-s + 71.6·22-s − 103.·23-s + 22.9·24-s − 48.5·26-s − 27·27-s − 216.·28-s + 108.·29-s − 147.·31-s − 223.·32-s − 57.5·33-s + 313.·34-s + ⋯ |
| L(s) = 1 | + 1.32·2-s − 0.577·3-s + 0.744·4-s − 0.762·6-s − 1.96·7-s − 0.337·8-s + 0.333·9-s + 0.526·11-s − 0.429·12-s − 0.277·13-s − 2.59·14-s − 1.19·16-s + 1.19·17-s + 0.440·18-s + 0.565·19-s + 1.13·21-s + 0.694·22-s − 0.941·23-s + 0.195·24-s − 0.366·26-s − 0.192·27-s − 1.46·28-s + 0.693·29-s − 0.854·31-s − 1.23·32-s − 0.303·33-s + 1.58·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.162063874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.162063874\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 - 3.73T + 8T^{2} \) |
| 7 | \( 1 + 36.4T + 343T^{2} \) |
| 11 | \( 1 - 19.1T + 1.33e3T^{2} \) |
| 17 | \( 1 - 83.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 46.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 103.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 108.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 147.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 160.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 231.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 340.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 119.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 732.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 229.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 108.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 10.3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 869.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 140.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 159.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.06e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 858.T + 9.12e5T^{2} \) |
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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
−9.687341674229537185438008853143, −9.113498919825187730729385535813, −7.54080538794698579330816600018, −6.66320891844764671386809973367, −5.98964960471724729511253818829, −5.45871368128765749224878360256, −4.16710169683622150056960208152, −3.52729812527958368136888090986, −2.61828313547544918889338055266, −0.64375560895884257902592056598,
0.64375560895884257902592056598, 2.61828313547544918889338055266, 3.52729812527958368136888090986, 4.16710169683622150056960208152, 5.45871368128765749224878360256, 5.98964960471724729511253818829, 6.66320891844764671386809973367, 7.54080538794698579330816600018, 9.113498919825187730729385535813, 9.687341674229537185438008853143
