| L(s) = 1 | + (−0.445 + 1.94i)2-s + (5.26 − 6.60i)3-s + (−3.60 − 1.73i)4-s + (5.40 − 6.77i)5-s + (10.5 + 13.2i)6-s + (−1.27 + 18.4i)7-s + (4.98 − 6.25i)8-s + (−9.85 − 43.1i)9-s + (10.8 + 13.5i)10-s + (13.1 − 57.5i)11-s + (−30.4 + 14.6i)12-s + (1.36 − 5.99i)13-s + (−35.4 − 10.7i)14-s + (−16.2 − 71.3i)15-s + (9.97 + 12.5i)16-s + (58.1 − 28.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.157 + 0.689i)2-s + (1.01 − 1.27i)3-s + (−0.450 − 0.216i)4-s + (0.483 − 0.606i)5-s + (0.716 + 0.898i)6-s + (−0.0686 + 0.997i)7-s + (0.220 − 0.276i)8-s + (−0.365 − 1.59i)9-s + (0.341 + 0.428i)10-s + (0.359 − 1.57i)11-s + (−0.732 + 0.352i)12-s + (0.0292 − 0.127i)13-s + (−0.676 − 0.204i)14-s + (−0.280 − 1.22i)15-s + (0.155 + 0.195i)16-s + (0.830 − 0.399i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.83175 - 0.751430i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.83175 - 0.751430i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.445 - 1.94i)T \) |
| 7 | \( 1 + (1.27 - 18.4i)T \) |
| good | 3 | \( 1 + (-5.26 + 6.60i)T + (-6.00 - 26.3i)T^{2} \) |
| 5 | \( 1 + (-5.40 + 6.77i)T + (-27.8 - 121. i)T^{2} \) |
| 11 | \( 1 + (-13.1 + 57.5i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (-1.36 + 5.99i)T + (-1.97e3 - 953. i)T^{2} \) |
| 17 | \( 1 + (-58.1 + 28.0i)T + (3.06e3 - 3.84e3i)T^{2} \) |
| 19 | \( 1 + 119.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-160. - 77.2i)T + (7.58e3 + 9.51e3i)T^{2} \) |
| 29 | \( 1 + (179. - 86.5i)T + (1.52e4 - 1.90e4i)T^{2} \) |
| 31 | \( 1 - 88.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-244. + 117. i)T + (3.15e4 - 3.96e4i)T^{2} \) |
| 41 | \( 1 + (295. - 370. i)T + (-1.53e4 - 6.71e4i)T^{2} \) |
| 43 | \( 1 + (-79.6 - 99.8i)T + (-1.76e4 + 7.75e4i)T^{2} \) |
| 47 | \( 1 + (75.7 - 332. i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (7.94 + 3.82i)T + (9.28e4 + 1.16e5i)T^{2} \) |
| 59 | \( 1 + (-243. - 305. i)T + (-4.57e4 + 2.00e5i)T^{2} \) |
| 61 | \( 1 + (-14.1 + 6.83i)T + (1.41e5 - 1.77e5i)T^{2} \) |
| 67 | \( 1 - 150.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-443. - 213. i)T + (2.23e5 + 2.79e5i)T^{2} \) |
| 73 | \( 1 + (14.7 + 64.6i)T + (-3.50e5 + 1.68e5i)T^{2} \) |
| 79 | \( 1 + 485.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (56.5 + 247. i)T + (-5.15e5 + 2.48e5i)T^{2} \) |
| 89 | \( 1 + (256. + 1.12e3i)T + (-6.35e5 + 3.05e5i)T^{2} \) |
| 97 | \( 1 + 127.T + 9.12e5T^{2} \) |
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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
−13.18788391763536185221494780909, −12.88315308457178602716795579877, −11.39954057036349467205858953053, −9.291012222052776047090552730653, −8.747331908000371475259417848903, −7.87976349438628833687342727031, −6.49136071766726910373146672723, −5.50556760607566434595545821129, −3.03341310280675091231829864085, −1.25783910393642238147299989786,
2.25590781940076925879691719454, 3.72930171704933860898269896029, 4.62896959422971720239257229899, 6.94720415506223853307373321992, 8.401822562100692301039869923512, 9.641791223699486450201548947672, 10.16497652105733240932658070448, 10.90277838725462228908934960732, 12.61260252687494106684342926200, 13.71605852614028029260529509649
