| L(s) = 1 | + (−1.33 − 4.10i)2-s + (7.83 + 2.54i)3-s + (−2.15 + 1.56i)4-s + 21.7·5-s − 35.5i·6-s + (7.98 − 5.80i)7-s + (−46.6 − 33.8i)8-s + (−10.6 − 7.75i)9-s + (−29.0 − 89.3i)10-s + (−4.48 − 6.16i)11-s + (−20.8 + 6.76i)12-s + (116. + 37.9i)13-s + (−34.4 − 25.0i)14-s + (170. + 55.3i)15-s + (−90.0 + 277. i)16-s + (−123. + 170. i)17-s + ⋯ |
| L(s) = 1 | + (−0.333 − 1.02i)2-s + (0.870 + 0.282i)3-s + (−0.134 + 0.0977i)4-s + 0.870·5-s − 0.988i·6-s + (0.162 − 0.118i)7-s + (−0.728 − 0.529i)8-s + (−0.131 − 0.0957i)9-s + (−0.290 − 0.893i)10-s + (−0.0370 − 0.0509i)11-s + (−0.144 + 0.0470i)12-s + (0.690 + 0.224i)13-s + (−0.175 − 0.127i)14-s + (0.757 + 0.246i)15-s + (−0.351 + 1.08i)16-s + (−0.427 + 0.588i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.278 + 0.960i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.278 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.35976 - 1.02196i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35976 - 1.02196i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + (628. - 726. i)T \) |
| good | 2 | \( 1 + (1.33 + 4.10i)T + (-12.9 + 9.40i)T^{2} \) |
| 3 | \( 1 + (-7.83 - 2.54i)T + (65.5 + 47.6i)T^{2} \) |
| 5 | \( 1 - 21.7T + 625T^{2} \) |
| 7 | \( 1 + (-7.98 + 5.80i)T + (741. - 2.28e3i)T^{2} \) |
| 11 | \( 1 + (4.48 + 6.16i)T + (-4.52e3 + 1.39e4i)T^{2} \) |
| 13 | \( 1 + (-116. - 37.9i)T + (2.31e4 + 1.67e4i)T^{2} \) |
| 17 | \( 1 + (123. - 170. i)T + (-2.58e4 - 7.94e4i)T^{2} \) |
| 19 | \( 1 + (-143. - 441. i)T + (-1.05e5 + 7.66e4i)T^{2} \) |
| 23 | \( 1 + (20.4 - 28.1i)T + (-8.64e4 - 2.66e5i)T^{2} \) |
| 29 | \( 1 + (-596. + 193. i)T + (5.72e5 - 4.15e5i)T^{2} \) |
| 37 | \( 1 - 1.79e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (634. + 1.95e3i)T + (-2.28e6 + 1.66e6i)T^{2} \) |
| 43 | \( 1 + (-38.7 + 12.6i)T + (2.76e6 - 2.00e6i)T^{2} \) |
| 47 | \( 1 + (-136. + 419. i)T + (-3.94e6 - 2.86e6i)T^{2} \) |
| 53 | \( 1 + (-1.57e3 + 2.16e3i)T + (-2.43e6 - 7.50e6i)T^{2} \) |
| 59 | \( 1 + (143. - 441. i)T + (-9.80e6 - 7.12e6i)T^{2} \) |
| 61 | \( 1 + 4.96e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 6.40e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + (4.72e3 + 3.43e3i)T + (7.85e6 + 2.41e7i)T^{2} \) |
| 73 | \( 1 + (5.09e3 + 7.01e3i)T + (-8.77e6 + 2.70e7i)T^{2} \) |
| 79 | \( 1 + (-1.43e3 + 1.98e3i)T + (-1.20e7 - 3.70e7i)T^{2} \) |
| 83 | \( 1 + (-7.08e3 + 2.30e3i)T + (3.83e7 - 2.78e7i)T^{2} \) |
| 89 | \( 1 + (5.88e3 + 8.09e3i)T + (-1.93e7 + 5.96e7i)T^{2} \) |
| 97 | \( 1 + (5.30e3 - 3.85e3i)T + (2.73e7 - 8.41e7i)T^{2} \) |
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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
−15.75704401636519202731115325336, −14.49592083309317327736106231759, −13.47140764457019623325977217717, −11.98889575790517399562045311362, −10.57978908454147543647611765820, −9.601499485780804498221480014661, −8.496858385337262857195978197139, −6.14251071591026133673581633430, −3.47986995741671159991473232336, −1.82023440230709000909929159331,
2.56119058695980546543958811701, 5.61330482987361321361405498015, 7.09866987987688340976285309282, 8.402733818412360354408954578819, 9.353028095579098208285475518494, 11.32643304673427557463925143965, 13.23930003136813688256242346046, 14.12062084533466927094342873307, 15.21876062744465597759404019300, 16.33432446313044691884112076799
