| L(s) = 1 | + (3.65 − 2.65i)2-s + (7.08 + 21.8i)3-s + (−3.58 + 11.0i)4-s + (−20.2 − 14.6i)5-s + (83.7 + 60.8i)6-s + (17.4 − 53.6i)7-s + (60.8 + 187. i)8-s + (−228. + 166. i)9-s − 112.·10-s + (59.3 + 396. i)11-s − 265.·12-s + (−694. + 504. i)13-s + (−78.7 − 242. i)14-s + (177. − 545. i)15-s + (419. + 304. i)16-s + (26.2 + 19.1i)17-s + ⋯ |
| L(s) = 1 | + (0.646 − 0.469i)2-s + (0.454 + 1.39i)3-s + (−0.111 + 0.344i)4-s + (−0.361 − 0.262i)5-s + (0.950 + 0.690i)6-s + (0.134 − 0.413i)7-s + (0.336 + 1.03i)8-s + (−0.940 + 0.683i)9-s − 0.357·10-s + (0.147 + 0.988i)11-s − 0.532·12-s + (−1.13 + 0.828i)13-s + (−0.107 − 0.330i)14-s + (0.203 − 0.625i)15-s + (0.409 + 0.297i)16-s + (0.0220 + 0.0160i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.50125 + 1.70117i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50125 + 1.70117i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (20.2 + 14.6i)T \) |
| 11 | \( 1 + (-59.3 - 396. i)T \) |
| good | 2 | \( 1 + (-3.65 + 2.65i)T + (9.88 - 30.4i)T^{2} \) |
| 3 | \( 1 + (-7.08 - 21.8i)T + (-196. + 142. i)T^{2} \) |
| 7 | \( 1 + (-17.4 + 53.6i)T + (-1.35e4 - 9.87e3i)T^{2} \) |
| 13 | \( 1 + (694. - 504. i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-26.2 - 19.1i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (208. + 641. i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 - 4.75e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (-2.08e3 + 6.42e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-5.60e3 + 4.07e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (774. - 2.38e3i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-2.52e3 - 7.77e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 - 1.07e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (2.83e3 + 8.73e3i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-516. + 375. i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (1.40e4 - 4.32e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (2.20e4 + 1.60e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 - 1.69e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (3.22e4 + 2.34e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (2.42e3 - 7.45e3i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (7.81e3 - 5.67e3i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (5.50e4 + 3.99e4i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 - 4.77e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-1.18e5 + 8.59e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−14.69187238189700597385087787185, −13.54823886778361005547717884611, −12.27108168278431752034657491569, −11.24658695288269435666651595196, −9.929678416685740574749354519164, −8.913412778708834262093305531359, −7.43505669921176845119996725221, −4.66072033452231118215772541805, −4.36423402720163003911403678129, −2.74124125991329825431008983731,
0.928532360516324527543679313533, 2.97251456660567158912452946173, 5.20258983089375220981513787832, 6.56495621650900262699677658845, 7.53418074372086954221760157071, 8.837130127528019042185844914273, 10.63891170109827213655015598559, 12.24490153625252126070285934484, 12.98622676667144066915176235255, 14.10423710788120928430751109382
