| L(s) = 1 | + (0.694 + 3.93i)2-s + (4.40 − 25.0i)3-s + (−15.0 + 5.47i)4-s + (68.9 + 57.8i)5-s + 101.·6-s + (55.3 + 46.4i)7-s + (−32 − 55.4i)8-s + (−377. − 137. i)9-s + (−180. + 311. i)10-s + (9.90 + 17.1i)11-s + (70.5 + 400. i)12-s + (615. − 224. i)13-s + (−144. + 250. i)14-s + (1.75e3 − 1.46e3i)15-s + (196. − 164. i)16-s + (1.44e3 + 527. i)17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (0.282 − 1.60i)3-s + (−0.469 + 0.171i)4-s + (1.23 + 1.03i)5-s + 1.15·6-s + (0.427 + 0.358i)7-s + (−0.176 − 0.306i)8-s + (−1.55 − 0.565i)9-s + (−0.569 + 0.986i)10-s + (0.0246 + 0.0427i)11-s + (0.141 + 0.801i)12-s + (1.01 − 0.367i)13-s + (−0.197 + 0.341i)14-s + (2.00 − 1.68i)15-s + (0.191 − 0.160i)16-s + (1.21 + 0.442i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0542i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0542i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.54429 + 0.0690491i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.54429 + 0.0690491i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.694 - 3.93i)T \) |
| 37 | \( 1 + (7.13e3 - 4.29e3i)T \) |
| good | 3 | \( 1 + (-4.40 + 25.0i)T + (-228. - 83.1i)T^{2} \) |
| 5 | \( 1 + (-68.9 - 57.8i)T + (542. + 3.07e3i)T^{2} \) |
| 7 | \( 1 + (-55.3 - 46.4i)T + (2.91e3 + 1.65e4i)T^{2} \) |
| 11 | \( 1 + (-9.90 - 17.1i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-615. + 224. i)T + (2.84e5 - 2.38e5i)T^{2} \) |
| 17 | \( 1 + (-1.44e3 - 527. i)T + (1.08e6 + 9.12e5i)T^{2} \) |
| 19 | \( 1 + (-150. + 851. i)T + (-2.32e6 - 8.46e5i)T^{2} \) |
| 23 | \( 1 + (-138. + 239. i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (3.18e3 + 5.51e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 - 6.58e3T + 2.86e7T^{2} \) |
| 41 | \( 1 + (1.71e4 - 6.24e3i)T + (8.87e7 - 7.44e7i)T^{2} \) |
| 43 | \( 1 - 1.85e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (802. - 1.39e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (2.29e4 - 1.92e4i)T + (7.26e7 - 4.11e8i)T^{2} \) |
| 59 | \( 1 + (1.63e4 - 1.37e4i)T + (1.24e8 - 7.04e8i)T^{2} \) |
| 61 | \( 1 + (1.05e4 - 3.83e3i)T + (6.46e8 - 5.42e8i)T^{2} \) |
| 67 | \( 1 + (-4.73e4 - 3.97e4i)T + (2.34e8 + 1.32e9i)T^{2} \) |
| 71 | \( 1 + (4.35e3 - 2.46e4i)T + (-1.69e9 - 6.17e8i)T^{2} \) |
| 73 | \( 1 + 5.39e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (6.63e4 + 5.56e4i)T + (5.34e8 + 3.03e9i)T^{2} \) |
| 83 | \( 1 + (6.62e4 + 2.41e4i)T + (3.01e9 + 2.53e9i)T^{2} \) |
| 89 | \( 1 + (-3.32e4 + 2.79e4i)T + (9.69e8 - 5.49e9i)T^{2} \) |
| 97 | \( 1 + (2.15e4 - 3.73e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
| 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
−13.71600359376558467969244209519, −12.95943020330754331036266463802, −11.64634487119011058054593825885, −10.12289886725869035615412146736, −8.583290343141421021806282564273, −7.54510868617711509770831591435, −6.41473053352862760820969270706, −5.75458756343951499749201780853, −2.87696792808420288194246948687, −1.44137247985691858031316220981,
1.44315318140054305317966478111, 3.47565621914210105256850985415, 4.76872384285148444843142634913, 5.63806420880504301860717736213, 8.479561813306758278297217530035, 9.355438227813645880856634008929, 10.09452474170751172382722933854, 11.03673408174087128469395787470, 12.47056812863600010490221041531, 13.84938157521243562502192201557
