| L(s) = 1 | + (0.394 + 1.34i)2-s + (1.97 − 1.70i)3-s + (0.0310 − 0.0199i)4-s + (−1.92 + 1.14i)5-s + (3.07 + 1.97i)6-s + (−2.68 − 1.22i)7-s + (2.15 + 1.86i)8-s + (0.541 − 3.76i)9-s + (−2.29 − 2.13i)10-s + (1.52 + 0.446i)11-s + (0.0271 − 0.0925i)12-s + (−4.24 + 1.93i)13-s + (0.587 − 4.08i)14-s + (−1.83 + 5.53i)15-s + (−1.63 + 3.57i)16-s + (0.714 − 1.11i)17-s + ⋯ |
| L(s) = 1 | + (0.279 + 0.950i)2-s + (1.13 − 0.986i)3-s + (0.0155 − 0.00999i)4-s + (−0.859 + 0.511i)5-s + (1.25 + 0.806i)6-s + (−1.01 − 0.462i)7-s + (0.762 + 0.660i)8-s + (0.180 − 1.25i)9-s + (−0.725 − 0.674i)10-s + (0.458 + 0.134i)11-s + (0.00784 − 0.0267i)12-s + (−1.17 + 0.537i)13-s + (0.157 − 1.09i)14-s + (−0.474 + 1.42i)15-s + (−0.407 + 0.892i)16-s + (0.173 − 0.269i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.43304 + 0.310455i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.43304 + 0.310455i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.92 - 1.14i)T \) |
| 23 | \( 1 + (-4.51 + 1.60i)T \) |
| good | 2 | \( 1 + (-0.394 - 1.34i)T + (-1.68 + 1.08i)T^{2} \) |
| 3 | \( 1 + (-1.97 + 1.70i)T + (0.426 - 2.96i)T^{2} \) |
| 7 | \( 1 + (2.68 + 1.22i)T + (4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (-1.52 - 0.446i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (4.24 - 1.93i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-0.714 + 1.11i)T + (-7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (4.70 - 3.02i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (4.78 + 3.07i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-4.02 + 4.64i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.226 - 0.0325i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (0.944 + 6.57i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-4.67 + 4.05i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 9.26iT - 47T^{2} \) |
| 53 | \( 1 + (-7.81 - 3.57i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-4.84 - 10.6i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (2.17 - 2.50i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (2.96 + 10.1i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (7.18 - 2.10i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (0.164 + 0.256i)T + (-30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (3.42 + 7.49i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (6.71 + 0.965i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-5.15 - 5.94i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-5.61 + 0.807i)T + (93.0 - 27.3i)T^{2} \) |
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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
−13.91909637054298898862344131401, −12.92391166569158446123805738009, −11.89648543773663104912387945477, −10.41534729227549018815095272477, −8.978774459540454287568038121754, −7.66061646526079204570735563345, −7.18838997866882536357892622976, −6.34618385123830999376806422160, −4.13311710664004643315947313224, −2.51389448811781161425533455624,
2.76883648969359807745187424919, 3.61137344505785783228506729316, 4.74956753433821405548390502852, 7.08561497398578711796461253641, 8.473177091924841557614084512739, 9.398796006932478382635832800281, 10.27938651898094476215791514713, 11.46766163252992852041140872841, 12.57819027243740651635996875935, 13.12719738468241323669993690679
