| L(s) = 1 | − 0.229·2-s − 7.94·4-s − 7·7-s + 3.66·8-s − 51.0·11-s − 46.0·13-s + 1.60·14-s + 62.7·16-s − 72.7·17-s − 123.·19-s + 11.7·22-s − 156.·23-s + 10.5·26-s + 55.6·28-s + 191.·29-s − 116.·31-s − 43.7·32-s + 16.7·34-s − 83.1·37-s + 28.4·38-s + 466.·41-s − 422.·43-s + 405.·44-s + 35.8·46-s − 268.·47-s + 49·49-s + 366.·52-s + ⋯ |
| L(s) = 1 | − 0.0812·2-s − 0.993·4-s − 0.377·7-s + 0.161·8-s − 1.39·11-s − 0.983·13-s + 0.0307·14-s + 0.980·16-s − 1.03·17-s − 1.49·19-s + 0.113·22-s − 1.41·23-s + 0.0798·26-s + 0.375·28-s + 1.22·29-s − 0.673·31-s − 0.241·32-s + 0.0843·34-s − 0.369·37-s + 0.121·38-s + 1.77·41-s − 1.49·43-s + 1.38·44-s + 0.114·46-s − 0.833·47-s + 0.142·49-s + 0.976·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1058441479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1058441479\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 0.229T + 8T^{2} \) |
| 11 | \( 1 + 51.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 72.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 123.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 156.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 191.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 116.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 83.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 466.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 422.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 268.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 310.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 709.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 402.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 114.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 214.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 402.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.37e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.15e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 366.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.06e3T + 9.12e5T^{2} \) |
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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
−9.002596179961646441774252652317, −8.303905380052380792863022501239, −7.68506390669204523716988045830, −6.60813552779326735178700689532, −5.71715160018101302709567071351, −4.75564771135651540192230181363, −4.24380110704325178450854776295, −2.95952022499671244712120702695, −2.00204214870745338140172871729, −0.15033813342853076249297006314,
0.15033813342853076249297006314, 2.00204214870745338140172871729, 2.95952022499671244712120702695, 4.24380110704325178450854776295, 4.75564771135651540192230181363, 5.71715160018101302709567071351, 6.60813552779326735178700689532, 7.68506390669204523716988045830, 8.303905380052380792863022501239, 9.002596179961646441774252652317
