| L(s) = 1 | + (−0.347 + 1.96i)2-s + (−1.43 − 8.12i)3-s + (−3.75 − 1.36i)4-s + (−2.21 + 1.86i)5-s + 16.5·6-s + (−21.7 + 18.2i)7-s + (4 − 6.92i)8-s + (−38.6 + 14.0i)9-s + (−2.89 − 5.01i)10-s + (−14.7 + 25.6i)11-s + (−5.73 + 32.5i)12-s + (−35.3 − 12.8i)13-s + (−28.4 − 49.2i)14-s + (18.3 + 15.3i)15-s + (12.2 + 10.2i)16-s + (31.2 − 11.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 + 0.696i)2-s + (−0.275 − 1.56i)3-s + (−0.469 − 0.171i)4-s + (−0.198 + 0.166i)5-s + 1.12·6-s + (−1.17 + 0.987i)7-s + (0.176 − 0.306i)8-s + (−1.43 + 0.521i)9-s + (−0.0916 − 0.158i)10-s + (−0.405 + 0.702i)11-s + (−0.137 + 0.782i)12-s + (−0.754 − 0.274i)13-s + (−0.542 − 0.940i)14-s + (0.315 + 0.264i)15-s + (0.191 + 0.160i)16-s + (0.445 − 0.162i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0159i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.000664008 - 0.0831546i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.000664008 - 0.0831546i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (-200. - 102. i)T \) |
| good | 3 | \( 1 + (1.43 + 8.12i)T + (-25.3 + 9.23i)T^{2} \) |
| 5 | \( 1 + (2.21 - 1.86i)T + (21.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (21.7 - 18.2i)T + (59.5 - 337. i)T^{2} \) |
| 11 | \( 1 + (14.7 - 25.6i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (35.3 + 12.8i)T + (1.68e3 + 1.41e3i)T^{2} \) |
| 17 | \( 1 + (-31.2 + 11.3i)T + (3.76e3 - 3.15e3i)T^{2} \) |
| 19 | \( 1 + (21.5 + 121. i)T + (-6.44e3 + 2.34e3i)T^{2} \) |
| 23 | \( 1 + (78.7 + 136. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (73.5 - 127. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 - 36.4T + 2.97e4T^{2} \) |
| 41 | \( 1 + (325. + 118. i)T + (5.27e4 + 4.43e4i)T^{2} \) |
| 43 | \( 1 - 160.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (271. + 469. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-3.05 - 2.55i)T + (2.58e4 + 1.46e5i)T^{2} \) |
| 59 | \( 1 + (104. + 87.5i)T + (3.56e4 + 2.02e5i)T^{2} \) |
| 61 | \( 1 + (-868. - 316. i)T + (1.73e5 + 1.45e5i)T^{2} \) |
| 67 | \( 1 + (209. - 175. i)T + (5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-118. - 670. i)T + (-3.36e5 + 1.22e5i)T^{2} \) |
| 73 | \( 1 + 537.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (100. - 84.1i)T + (8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (293. - 106. i)T + (4.38e5 - 3.67e5i)T^{2} \) |
| 89 | \( 1 + (250. + 210. i)T + (1.22e5 + 6.94e5i)T^{2} \) |
| 97 | \( 1 + (761. + 1.31e3i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
| 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
−13.20437760822088405131635069898, −12.71918138608557199025459943565, −11.76570209477769990689086199826, −9.922798238977339972850791803131, −8.596876963125803043145759848778, −7.28459380249184613953064172569, −6.61914021141386229286984834077, −5.36788921867924913118320964971, −2.55043259145738651873474776796, −0.05442594685603346558542267523,
3.39972391386602856958948536209, 4.25779999119246987390645699536, 5.86852171112289756753926768817, 7.977731302590716205250352796089, 9.714323596334814607906398201229, 9.948757304179144371815473919800, 11.01925666484032539259854515628, 12.20077861271558986453800116103, 13.49657119465086336227866058916, 14.61156961148591891056809398127
