| L(s) = 1 | + (0.347 − 1.96i)2-s + (0.551 + 3.12i)3-s + (−3.75 − 1.36i)4-s + (7.17 − 6.02i)5-s + 6.35·6-s + (7.01 − 5.88i)7-s + (−4 + 6.92i)8-s + (15.8 − 5.78i)9-s + (−9.36 − 16.2i)10-s + (21.0 − 36.4i)11-s + (2.20 − 12.5i)12-s + (47.5 + 17.3i)13-s + (−9.16 − 15.8i)14-s + (22.7 + 19.1i)15-s + (12.2 + 10.2i)16-s + (−120. + 43.8i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (0.106 + 0.602i)3-s + (−0.469 − 0.171i)4-s + (0.641 − 0.538i)5-s + 0.432·6-s + (0.378 − 0.318i)7-s + (−0.176 + 0.306i)8-s + (0.588 − 0.214i)9-s + (−0.296 − 0.513i)10-s + (0.577 − 0.999i)11-s + (0.0530 − 0.301i)12-s + (1.01 + 0.369i)13-s + (−0.174 − 0.302i)14-s + (0.392 + 0.329i)15-s + (0.191 + 0.160i)16-s + (−1.71 + 0.624i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.579 + 0.815i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.579 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.60466 - 0.828493i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.60466 - 0.828493i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.347 + 1.96i)T \) |
| 37 | \( 1 + (-29.1 - 223. i)T \) |
| good | 3 | \( 1 + (-0.551 - 3.12i)T + (-25.3 + 9.23i)T^{2} \) |
| 5 | \( 1 + (-7.17 + 6.02i)T + (21.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (-7.01 + 5.88i)T + (59.5 - 337. i)T^{2} \) |
| 11 | \( 1 + (-21.0 + 36.4i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-47.5 - 17.3i)T + (1.68e3 + 1.41e3i)T^{2} \) |
| 17 | \( 1 + (120. - 43.8i)T + (3.76e3 - 3.15e3i)T^{2} \) |
| 19 | \( 1 + (19.9 + 113. i)T + (-6.44e3 + 2.34e3i)T^{2} \) |
| 23 | \( 1 + (-51.0 - 88.3i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (61.2 - 106. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + 59.7T + 2.97e4T^{2} \) |
| 41 | \( 1 + (76.1 + 27.7i)T + (5.27e4 + 4.43e4i)T^{2} \) |
| 43 | \( 1 - 39.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + (162. + 281. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (400. + 336. i)T + (2.58e4 + 1.46e5i)T^{2} \) |
| 59 | \( 1 + (-135. - 113. i)T + (3.56e4 + 2.02e5i)T^{2} \) |
| 61 | \( 1 + (-195. - 71.2i)T + (1.73e5 + 1.45e5i)T^{2} \) |
| 67 | \( 1 + (551. - 462. i)T + (5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (110. + 626. i)T + (-3.36e5 + 1.22e5i)T^{2} \) |
| 73 | \( 1 - 578.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (36.6 - 30.7i)T + (8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-718. + 261. i)T + (4.38e5 - 3.67e5i)T^{2} \) |
| 89 | \( 1 + (124. + 104. i)T + (1.22e5 + 6.94e5i)T^{2} \) |
| 97 | \( 1 + (-407. - 705. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−13.46463535221561348994119411113, −13.19003301602620371383891635282, −11.37706915289579360441246850827, −10.76796725802982716045365323455, −9.270860884502161029683376802379, −8.774604419228113007959355473114, −6.54139695112353471015743982704, −4.89786197047319955073353782535, −3.70567241672580143915932962666, −1.44018420698902136421200125666,
1.99622247036476750767917685921, 4.39252133795219522540164337811, 6.13155788343747896201021758079, 6.99235771364680633512669649699, 8.258844482913569114770793526862, 9.564271057418301565111674785975, 10.86413324590856497758501539774, 12.40751993653437621359260689684, 13.31377014204792955927641471174, 14.24714759449038731531225363644
