| L(s) = 1 | + (0.356 − 0.781i)2-s + (5.20 + 1.52i)3-s + (4.75 + 5.48i)4-s + (4.20 − 2.70i)5-s + (3.05 − 3.52i)6-s + (3.53 − 24.5i)7-s + (12.5 − 3.69i)8-s + (2.07 + 1.33i)9-s + (−0.611 − 4.25i)10-s + (0.0443 + 0.0970i)11-s + (16.3 + 35.8i)12-s + (12.1 + 84.7i)13-s + (−17.9 − 11.5i)14-s + (26.0 − 7.64i)15-s + (−6.66 + 46.3i)16-s + (38.7 − 44.7i)17-s + ⋯ |
| L(s) = 1 | + (0.126 − 0.276i)2-s + (1.00 + 0.294i)3-s + (0.594 + 0.686i)4-s + (0.376 − 0.241i)5-s + (0.207 − 0.239i)6-s + (0.190 − 1.32i)7-s + (0.556 − 0.163i)8-s + (0.0769 + 0.0494i)9-s + (−0.0193 − 0.134i)10-s + (0.00121 + 0.00266i)11-s + (0.393 + 0.862i)12-s + (0.260 + 1.80i)13-s + (−0.342 − 0.220i)14-s + (0.448 − 0.131i)15-s + (−0.104 + 0.724i)16-s + (0.553 − 0.638i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0604i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0604i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.75506 - 0.0833290i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.75506 - 0.0833290i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-4.20 + 2.70i)T \) |
| 23 | \( 1 + (54.2 - 96.0i)T \) |
| good | 2 | \( 1 + (-0.356 + 0.781i)T + (-5.23 - 6.04i)T^{2} \) |
| 3 | \( 1 + (-5.20 - 1.52i)T + (22.7 + 14.5i)T^{2} \) |
| 7 | \( 1 + (-3.53 + 24.5i)T + (-329. - 96.6i)T^{2} \) |
| 11 | \( 1 + (-0.0443 - 0.0970i)T + (-871. + 1.00e3i)T^{2} \) |
| 13 | \( 1 + (-12.1 - 84.7i)T + (-2.10e3 + 618. i)T^{2} \) |
| 17 | \( 1 + (-38.7 + 44.7i)T + (-699. - 4.86e3i)T^{2} \) |
| 19 | \( 1 + (48.9 + 56.4i)T + (-976. + 6.78e3i)T^{2} \) |
| 29 | \( 1 + (26.6 - 30.7i)T + (-3.47e3 - 2.41e4i)T^{2} \) |
| 31 | \( 1 + (33.2 - 9.76i)T + (2.50e4 - 1.61e4i)T^{2} \) |
| 37 | \( 1 + (-21.6 - 13.9i)T + (2.10e4 + 4.60e4i)T^{2} \) |
| 41 | \( 1 + (20.3 - 13.1i)T + (2.86e4 - 6.26e4i)T^{2} \) |
| 43 | \( 1 + (310. + 91.0i)T + (6.68e4 + 4.29e4i)T^{2} \) |
| 47 | \( 1 + 159.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-22.4 + 156. i)T + (-1.42e5 - 4.19e4i)T^{2} \) |
| 59 | \( 1 + (22.1 + 154. i)T + (-1.97e5 + 5.78e4i)T^{2} \) |
| 61 | \( 1 + (262. - 77.1i)T + (1.90e5 - 1.22e5i)T^{2} \) |
| 67 | \( 1 + (390. - 854. i)T + (-1.96e5 - 2.27e5i)T^{2} \) |
| 71 | \( 1 + (-438. + 960. i)T + (-2.34e5 - 2.70e5i)T^{2} \) |
| 73 | \( 1 + (374. + 432. i)T + (-5.53e4 + 3.85e5i)T^{2} \) |
| 79 | \( 1 + (40.0 + 278. i)T + (-4.73e5 + 1.38e5i)T^{2} \) |
| 83 | \( 1 + (-881. - 566. i)T + (2.37e5 + 5.20e5i)T^{2} \) |
| 89 | \( 1 + (-311. - 91.3i)T + (5.93e5 + 3.81e5i)T^{2} \) |
| 97 | \( 1 + (1.22e3 - 785. i)T + (3.79e5 - 8.30e5i)T^{2} \) |
| 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
−13.49366034527918865278081155542, −11.95181224708811988565160981248, −11.08261801712774574994617851275, −9.820547961137881367431945331776, −8.812425512185710479877285602241, −7.64133417636684675381539006803, −6.67153429725209166087257137335, −4.40408953562626374435821753905, −3.43140026011226507848683902207, −1.85765090792408346945968273283,
1.91484403174501758563525884937, 2.98822537418689179472290777137, 5.44646495711247306966170528873, 6.15847770928572394405070686865, 7.83158944485687262733040294028, 8.531484301947725098944190899226, 9.935515107457423189976028890266, 10.85256723978772520699102652706, 12.25348734505018241002561246957, 13.23859367380046964220754867927
