| L(s) = 1 | + (−1.86 − 1.95i)2-s + (−1.83 − 0.689i)3-s + (−0.238 + 5.31i)4-s + (−1.14 − 0.829i)5-s + (2.08 + 4.87i)6-s + (−0.196 + 1.44i)7-s + (6.76 − 5.91i)8-s + (0.636 + 0.556i)9-s + (0.512 + 3.78i)10-s + (−4.77 + 2.85i)11-s + (4.10 − 9.59i)12-s + (−3.98 − 2.38i)13-s + (3.20 − 2.32i)14-s + (1.52 + 2.31i)15-s + (−13.6 − 1.22i)16-s + (0.776 − 2.38i)17-s + ⋯ |
| L(s) = 1 | + (−1.32 − 1.38i)2-s + (−1.05 − 0.397i)3-s + (−0.119 + 2.65i)4-s + (−0.510 − 0.371i)5-s + (0.851 + 1.99i)6-s + (−0.0741 + 0.547i)7-s + (2.39 − 2.08i)8-s + (0.212 + 0.185i)9-s + (0.162 + 1.19i)10-s + (−1.43 + 0.859i)11-s + (1.18 − 2.76i)12-s + (−1.10 − 0.660i)13-s + (0.855 − 0.621i)14-s + (0.393 + 0.596i)15-s + (−3.40 − 0.306i)16-s + (0.188 − 0.579i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.389 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.389 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0391497 + 0.0590578i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0391497 + 0.0590578i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 + (8.11 + 2.26i)T \) |
| good | 2 | \( 1 + (1.86 + 1.95i)T + (-0.0897 + 1.99i)T^{2} \) |
| 3 | \( 1 + (1.83 + 0.689i)T + (2.25 + 1.97i)T^{2} \) |
| 5 | \( 1 + (1.14 + 0.829i)T + (1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (0.196 - 1.44i)T + (-6.74 - 1.86i)T^{2} \) |
| 11 | \( 1 + (4.77 - 2.85i)T + (5.21 - 9.68i)T^{2} \) |
| 13 | \( 1 + (3.98 + 2.38i)T + (6.16 + 11.4i)T^{2} \) |
| 17 | \( 1 + (-0.776 + 2.38i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.17 + 1.77i)T + (-7.46 - 17.4i)T^{2} \) |
| 23 | \( 1 + (2.12 + 1.02i)T + (14.3 + 17.9i)T^{2} \) |
| 29 | \( 1 + (1.14 - 0.314i)T + (24.8 - 14.8i)T^{2} \) |
| 31 | \( 1 + (-4.65 + 0.418i)T + (30.5 - 5.53i)T^{2} \) |
| 37 | \( 1 + (-0.278 + 0.134i)T + (23.0 - 28.9i)T^{2} \) |
| 41 | \( 1 + (-0.532 + 2.33i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (11.2 + 2.04i)T + (40.2 + 15.1i)T^{2} \) |
| 47 | \( 1 + (1.41 - 0.529i)T + (35.3 - 30.9i)T^{2} \) |
| 53 | \( 1 + (0.0346 + 0.770i)T + (-52.7 + 4.75i)T^{2} \) |
| 59 | \( 1 + (0.617 - 1.44i)T + (-40.7 - 42.6i)T^{2} \) |
| 61 | \( 1 + (0.727 + 5.37i)T + (-58.8 + 16.2i)T^{2} \) |
| 67 | \( 1 + (0.221 - 4.92i)T + (-66.7 - 6.00i)T^{2} \) |
| 73 | \( 1 + (-3.71 - 3.88i)T + (-3.27 + 72.9i)T^{2} \) |
| 79 | \( 1 + (0.421 - 0.368i)T + (10.6 - 78.2i)T^{2} \) |
| 83 | \( 1 + (-0.805 + 1.88i)T + (-57.3 - 59.9i)T^{2} \) |
| 89 | \( 1 + (-0.320 - 7.14i)T + (-88.6 + 7.97i)T^{2} \) |
| 97 | \( 1 + (-2.18 - 9.55i)T + (-87.3 + 42.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.05574586422541914180604062016, −12.20874321175025473790732754777, −11.84421773709624519982793013707, −10.57240395581690610409366866350, −9.721781728753142141676805124471, −8.258349628654983194406659954580, −7.29245765302360382599474946767, −4.98270941824056791767320799333, −2.61743573873163801737323285501, −0.15537322009243603180386293444,
4.99659416335435403743600351846, 6.06971467324191798830744915258, 7.35356468288640654747373366937, 8.229793500415359188686420631042, 9.921796500320373817429899965223, 10.56345741072964828162223737137, 11.56496847556830757316640986363, 13.70409858327796816107114457598, 14.90889796155663723222009269730, 15.83294230360091033524562386723
