| L(s) = 1 | + (2.11 − 1.96i)2-s + (24.5 − 3.70i)3-s + (−1.76 + 23.5i)4-s + (4.41 − 11.2i)5-s + (44.7 − 56.1i)6-s + (97.0 + 85.9i)7-s + (100. + 125. i)8-s + (356. − 110. i)9-s + (−12.7 − 32.5i)10-s + (−589. − 181. i)11-s + (43.8 + 585. i)12-s + (−4.66 − 20.4i)13-s + (374. − 8.59i)14-s + (66.7 − 292. i)15-s + (−288. − 43.4i)16-s + (1.64e3 − 1.11e3i)17-s + ⋯ |
| L(s) = 1 | + (0.374 − 0.347i)2-s + (1.57 − 0.237i)3-s + (−0.0552 + 0.736i)4-s + (0.0789 − 0.201i)5-s + (0.507 − 0.636i)6-s + (0.748 + 0.663i)7-s + (0.554 + 0.694i)8-s + (1.46 − 0.453i)9-s + (−0.0403 − 0.102i)10-s + (−1.46 − 0.453i)11-s + (0.0879 + 1.17i)12-s + (−0.00765 − 0.0335i)13-s + (0.510 − 0.0117i)14-s + (0.0766 − 0.335i)15-s + (−0.281 − 0.0424i)16-s + (1.37 − 0.939i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0609i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0609i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.25437 - 0.0992813i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.25437 - 0.0992813i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-97.0 - 85.9i)T \) |
| good | 2 | \( 1 + (-2.11 + 1.96i)T + (2.39 - 31.9i)T^{2} \) |
| 3 | \( 1 + (-24.5 + 3.70i)T + (232. - 71.6i)T^{2} \) |
| 5 | \( 1 + (-4.41 + 11.2i)T + (-2.29e3 - 2.12e3i)T^{2} \) |
| 11 | \( 1 + (589. + 181. i)T + (1.33e5 + 9.07e4i)T^{2} \) |
| 13 | \( 1 + (4.66 + 20.4i)T + (-3.34e5 + 1.61e5i)T^{2} \) |
| 17 | \( 1 + (-1.64e3 + 1.11e3i)T + (5.18e5 - 1.32e6i)T^{2} \) |
| 19 | \( 1 + (361. + 626. i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (1.88e3 + 1.28e3i)T + (2.35e6 + 5.99e6i)T^{2} \) |
| 29 | \( 1 + (7.20e3 + 3.46e3i)T + (1.27e7 + 1.60e7i)T^{2} \) |
| 31 | \( 1 + (-1.74e3 + 3.02e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-387. - 5.17e3i)T + (-6.85e7 + 1.03e7i)T^{2} \) |
| 41 | \( 1 + (242. + 303. i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 43 | \( 1 + (8.20e3 - 1.02e4i)T + (-3.27e7 - 1.43e8i)T^{2} \) |
| 47 | \( 1 + (1.46e4 - 1.36e4i)T + (1.71e7 - 2.28e8i)T^{2} \) |
| 53 | \( 1 + (-667. + 8.91e3i)T + (-4.13e8 - 6.23e7i)T^{2} \) |
| 59 | \( 1 + (-91.4 - 232. i)T + (-5.24e8 + 4.86e8i)T^{2} \) |
| 61 | \( 1 + (853. + 1.13e4i)T + (-8.35e8 + 1.25e8i)T^{2} \) |
| 67 | \( 1 + (-3.50e4 + 6.06e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-4.17e4 + 2.01e4i)T + (1.12e9 - 1.41e9i)T^{2} \) |
| 73 | \( 1 + (-2.39e4 - 2.22e4i)T + (1.54e8 + 2.06e9i)T^{2} \) |
| 79 | \( 1 + (-2.79e4 - 4.83e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-3.18e3 + 1.39e4i)T + (-3.54e9 - 1.70e9i)T^{2} \) |
| 89 | \( 1 + (-7.41e4 + 2.28e4i)T + (4.61e9 - 3.14e9i)T^{2} \) |
| 97 | \( 1 + 1.05e5T + 8.58e9T^{2} \) |
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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
−14.33776933247031339386428455532, −13.42765818488397154498736972041, −12.63470759166584146384982509499, −11.29815383633224672047251706068, −9.479381142388296098865841495074, −8.138399335337496592155206028148, −7.80979121114429049579388778225, −5.04533521781161826468248013792, −3.23210023035917476834870551596, −2.24045873863637186316524980715,
1.84880468494928771206298201010, 3.77151864168311954284214275128, 5.28801664911619196068647862786, 7.36018912114779109971364420327, 8.273560595616746422812516374051, 9.939365486771730395784610684523, 10.55048922019217178622821677518, 12.87216359143206770362252191531, 13.88457932851682450785301401517, 14.61508233301009603041558344151
