| L(s) = 1 | + (14.9 + 2.64i)3-s + (1.74 + 1.46i)5-s + (22.6 + 39.2i)7-s + (141. + 51.5i)9-s + (51.9 − 89.9i)11-s + (−220. + 38.9i)13-s + (22.3 + 26.6i)15-s + (−69.3 + 25.2i)17-s + (340. + 118. i)19-s + (235. + 647. i)21-s + (547. − 459. i)23-s + (−107. − 610. i)25-s + (918. + 530. i)27-s + (−295. + 813. i)29-s + (−723. + 417. i)31-s + ⋯ |
| L(s) = 1 | + (1.66 + 0.293i)3-s + (0.0699 + 0.0586i)5-s + (0.462 + 0.800i)7-s + (1.74 + 0.636i)9-s + (0.429 − 0.743i)11-s + (−1.30 + 0.230i)13-s + (0.0992 + 0.118i)15-s + (−0.239 + 0.0873i)17-s + (0.944 + 0.328i)19-s + (0.534 + 1.46i)21-s + (1.03 − 0.867i)23-s + (−0.172 − 0.976i)25-s + (1.26 + 0.727i)27-s + (−0.351 + 0.966i)29-s + (−0.753 + 0.434i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 - 0.465i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.885 - 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.76346 + 0.682165i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.76346 + 0.682165i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-340. - 118. i)T \) |
| good | 3 | \( 1 + (-14.9 - 2.64i)T + (76.1 + 27.7i)T^{2} \) |
| 5 | \( 1 + (-1.74 - 1.46i)T + (108. + 615. i)T^{2} \) |
| 7 | \( 1 + (-22.6 - 39.2i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-51.9 + 89.9i)T + (-7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (220. - 38.9i)T + (2.68e4 - 9.76e3i)T^{2} \) |
| 17 | \( 1 + (69.3 - 25.2i)T + (6.39e4 - 5.36e4i)T^{2} \) |
| 23 | \( 1 + (-547. + 459. i)T + (4.85e4 - 2.75e5i)T^{2} \) |
| 29 | \( 1 + (295. - 813. i)T + (-5.41e5 - 4.54e5i)T^{2} \) |
| 31 | \( 1 + (723. - 417. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + 603. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (1.04e3 + 183. i)T + (2.65e6 + 9.66e5i)T^{2} \) |
| 43 | \( 1 + (2.61e3 + 2.19e3i)T + (5.93e5 + 3.36e6i)T^{2} \) |
| 47 | \( 1 + (1.02e3 + 372. i)T + (3.73e6 + 3.13e6i)T^{2} \) |
| 53 | \( 1 + (1.04e3 + 1.24e3i)T + (-1.37e6 + 7.77e6i)T^{2} \) |
| 59 | \( 1 + (653. + 1.79e3i)T + (-9.28e6 + 7.78e6i)T^{2} \) |
| 61 | \( 1 + (-2.68e3 + 2.25e3i)T + (2.40e6 - 1.36e7i)T^{2} \) |
| 67 | \( 1 + (1.44e3 - 3.97e3i)T + (-1.54e7 - 1.29e7i)T^{2} \) |
| 71 | \( 1 + (3.64e3 - 4.34e3i)T + (-4.41e6 - 2.50e7i)T^{2} \) |
| 73 | \( 1 + (337. - 1.91e3i)T + (-2.66e7 - 9.71e6i)T^{2} \) |
| 79 | \( 1 + (6.05e3 + 1.06e3i)T + (3.66e7 + 1.33e7i)T^{2} \) |
| 83 | \( 1 + (-3.62e3 - 6.28e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-1.04e4 + 1.84e3i)T + (5.89e7 - 2.14e7i)T^{2} \) |
| 97 | \( 1 + (4.97e3 + 1.36e4i)T + (-6.78e7 + 5.69e7i)T^{2} \) |
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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
−14.34310487598645049137554991892, −12.97425368122304756792144555131, −11.75959199224505750865786244641, −10.17728825828138414890647222352, −9.052540337743955998403435037951, −8.419486984627607097853618769461, −7.09738898329044381039028243524, −5.04033779840967416421757791930, −3.37170180194869652357854658888, −2.11993587646237701322159836014,
1.64970565001316583867850162744, 3.20616952871512378155840686588, 4.70867720429921382756042244655, 7.23808008307181438359140046561, 7.67636510671905947428477720227, 9.203944080310662121501112186515, 9.873576279742943949785234009873, 11.58693096328914897605576725255, 13.03114376312127854166001368145, 13.69368023507081718669943537037
