| L(s) = 1 | + (3.75 + 1.36i)2-s + (0.403 + 2.29i)3-s + (12.2 + 10.2i)4-s + (−72.6 + 60.9i)5-s + (−1.61 + 9.16i)6-s + (−55.7 + 96.5i)7-s + (32.0 + 55.4i)8-s + (223. − 81.2i)9-s + (−356. + 129. i)10-s + (59.6 + 103. i)11-s + (−18.6 + 32.2i)12-s + (−70.3 + 399. i)13-s + (−341. + 286. i)14-s + (−168. − 141. i)15-s + (44.4 + 252. i)16-s + (−695. − 253. i)17-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (0.0259 + 0.146i)3-s + (0.383 + 0.321i)4-s + (−1.29 + 1.09i)5-s + (−0.0183 + 0.103i)6-s + (−0.430 + 0.745i)7-s + (0.176 + 0.306i)8-s + (0.918 − 0.334i)9-s + (−1.12 + 0.410i)10-s + (0.148 + 0.257i)11-s + (−0.0373 + 0.0646i)12-s + (−0.115 + 0.654i)13-s + (−0.466 + 0.391i)14-s + (−0.193 − 0.162i)15-s + (0.0434 + 0.246i)16-s + (−0.583 − 0.212i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.419 - 0.907i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.930560 + 1.45574i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.930560 + 1.45574i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.75 - 1.36i)T \) |
| 19 | \( 1 + (-1.07e3 + 1.15e3i)T \) |
| good | 3 | \( 1 + (-0.403 - 2.29i)T + (-228. + 83.1i)T^{2} \) |
| 5 | \( 1 + (72.6 - 60.9i)T + (542. - 3.07e3i)T^{2} \) |
| 7 | \( 1 + (55.7 - 96.5i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-59.6 - 103. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (70.3 - 399. i)T + (-3.48e5 - 1.26e5i)T^{2} \) |
| 17 | \( 1 + (695. + 253. i)T + (1.08e6 + 9.12e5i)T^{2} \) |
| 23 | \( 1 + (-2.58e3 - 2.17e3i)T + (1.11e6 + 6.33e6i)T^{2} \) |
| 29 | \( 1 + (588. - 214. i)T + (1.57e7 - 1.31e7i)T^{2} \) |
| 31 | \( 1 + (-1.97e3 + 3.41e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 - 1.37e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (781. + 4.43e3i)T + (-1.08e8 + 3.96e7i)T^{2} \) |
| 43 | \( 1 + (1.48e4 - 1.24e4i)T + (2.55e7 - 1.44e8i)T^{2} \) |
| 47 | \( 1 + (1.96e4 - 7.15e3i)T + (1.75e8 - 1.47e8i)T^{2} \) |
| 53 | \( 1 + (-2.99e4 - 2.50e4i)T + (7.26e7 + 4.11e8i)T^{2} \) |
| 59 | \( 1 + (1.77e4 + 6.46e3i)T + (5.47e8 + 4.59e8i)T^{2} \) |
| 61 | \( 1 + (-1.18e4 - 9.94e3i)T + (1.46e8 + 8.31e8i)T^{2} \) |
| 67 | \( 1 + (-9.27e3 + 3.37e3i)T + (1.03e9 - 8.67e8i)T^{2} \) |
| 71 | \( 1 + (-1.05e3 + 889. i)T + (3.13e8 - 1.77e9i)T^{2} \) |
| 73 | \( 1 + (1.49e4 + 8.46e4i)T + (-1.94e9 + 7.09e8i)T^{2} \) |
| 79 | \( 1 + (1.47e3 + 8.34e3i)T + (-2.89e9 + 1.05e9i)T^{2} \) |
| 83 | \( 1 + (-4.50e3 + 7.80e3i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (1.21e4 - 6.90e4i)T + (-5.24e9 - 1.90e9i)T^{2} \) |
| 97 | \( 1 + (-1.07e5 - 3.92e4i)T + (6.57e9 + 5.51e9i)T^{2} \) |
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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
−15.33441042910484273409120967051, −14.94166501406602117524857403934, −13.30120982392336251958006140770, −11.97457159401068864178555247842, −11.18506776691372175769824175200, −9.419866920970451049004916480629, −7.49753247855787131685790078922, −6.59119980936864891836079157753, −4.42184868310903699206242179099, −3.02731527283565954855704842782,
0.857085275279120920623689107173, 3.71711506626693147562884287346, 4.86491413354659866201651901935, 7.03260671124010045920820939978, 8.292414114971817353737526025363, 10.14352516888902164132467608476, 11.53084674586016477429220146998, 12.70955624637718107613287970703, 13.29853059959422390417083718873, 14.99801133293949815139172005719
