| L(s) = 1 | + (−2 + 3.46i)2-s + (−10.0 − 17.3i)3-s + (−7.99 − 13.8i)4-s + (−32.4 − 56.1i)5-s + 80.0·6-s + (−55.2 − 95.6i)7-s + 63.9·8-s + (−78.6 + 136. i)9-s + 259.·10-s − 26.5·11-s + (−160. + 277. i)12-s + (88.6 + 153. i)13-s + 441.·14-s + (−648. + 1.12e3i)15-s + (−128 + 221. i)16-s + (−371. + 642. i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.641 − 1.11i)3-s + (−0.249 − 0.433i)4-s + (−0.579 − 1.00i)5-s + 0.907·6-s + (−0.426 − 0.738i)7-s + 0.353·8-s + (−0.323 + 0.560i)9-s + 0.820·10-s − 0.0661·11-s + (−0.320 + 0.555i)12-s + (0.145 + 0.251i)13-s + 0.602·14-s + (−0.744 + 1.28i)15-s + (−0.125 + 0.216i)16-s + (−0.311 + 0.539i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.716 - 0.697i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0779640 + 0.191813i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0779640 + 0.191813i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2 - 3.46i)T \) |
| 37 | \( 1 + (-5.04e3 - 6.62e3i)T \) |
| good | 3 | \( 1 + (10.0 + 17.3i)T + (-121.5 + 210. i)T^{2} \) |
| 5 | \( 1 + (32.4 + 56.1i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (55.2 + 95.6i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + 26.5T + 1.61e5T^{2} \) |
| 13 | \( 1 + (-88.6 - 153. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + (371. - 642. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-16.1 - 27.9i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 - 59.3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 332.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.86e3T + 2.86e7T^{2} \) |
| 41 | \( 1 + (176. + 306. i)T + (-5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 - 9.17e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.96e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + (1.30e4 - 2.25e4i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (3.06e3 - 5.31e3i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-2.15e3 - 3.72e3i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (2.39e4 + 4.14e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (3.58e4 + 6.21e4i)T + (-9.02e8 + 1.56e9i)T^{2} \) |
| 73 | \( 1 + 3.44e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (4.23e4 + 7.33e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (4.33e4 - 7.50e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (7.11e4 - 1.23e5i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + 3.76e4T + 8.58e9T^{2} \) |
| 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
−12.89146654855423617101263319824, −12.04887578308641347013785744381, −10.74510550103998047326710746140, −9.181588465305471228558484971901, −7.951178597588452097903305433539, −7.01719559320025062065125405763, −5.89186431060461139770373755075, −4.29488747000441289392626624281, −1.24834374333585548732921422124, −0.12399109249345153597382941822,
2.79951733908432248989984285189, 4.09370747887354007950207875047, 5.67178888809866258136168536703, 7.32076540765396416480474548497, 8.979409036877657443458782878017, 10.03044547699907980521512914730, 10.96821285336812155671855901545, 11.58553057197758769588795341497, 12.86234258287702069152625734833, 14.49598189091387853097855838140
