| L(s) = 1 | + (−0.0282 − 0.196i)2-s + (7.63 − 2.24i)4-s + (6.59 − 7.60i)5-s + (21.3 − 13.7i)7-s + (−1.31 − 2.88i)8-s + (−1.68 − 1.08i)10-s + (−2.01 + 14.0i)11-s + (−3.85 − 2.47i)13-s + (−3.29 − 3.80i)14-s + (53.0 − 34.0i)16-s + (−98.0 − 28.7i)17-s + (−66.4 + 19.5i)19-s + (33.2 − 72.8i)20-s + 2.81·22-s + (83.9 + 71.5i)23-s + ⋯ |
| L(s) = 1 | + (−0.00999 − 0.0695i)2-s + (0.954 − 0.280i)4-s + (0.589 − 0.680i)5-s + (1.15 − 0.739i)7-s + (−0.0582 − 0.127i)8-s + (−0.0532 − 0.0341i)10-s + (−0.0552 + 0.384i)11-s + (−0.0822 − 0.0528i)13-s + (−0.0629 − 0.0726i)14-s + (0.828 − 0.532i)16-s + (−1.39 − 0.410i)17-s + (−0.802 + 0.235i)19-s + (0.372 − 0.815i)20-s + 0.0272·22-s + (0.761 + 0.648i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.23919 - 1.26420i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.23919 - 1.26420i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + (-83.9 - 71.5i)T \) |
| good | 2 | \( 1 + (0.0282 + 0.196i)T + (-7.67 + 2.25i)T^{2} \) |
| 5 | \( 1 + (-6.59 + 7.60i)T + (-17.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (-21.3 + 13.7i)T + (142. - 312. i)T^{2} \) |
| 11 | \( 1 + (2.01 - 14.0i)T + (-1.27e3 - 374. i)T^{2} \) |
| 13 | \( 1 + (3.85 + 2.47i)T + (912. + 1.99e3i)T^{2} \) |
| 17 | \( 1 + (98.0 + 28.7i)T + (4.13e3 + 2.65e3i)T^{2} \) |
| 19 | \( 1 + (66.4 - 19.5i)T + (5.77e3 - 3.70e3i)T^{2} \) |
| 29 | \( 1 + (-202. - 59.4i)T + (2.05e4 + 1.31e4i)T^{2} \) |
| 31 | \( 1 + (117. + 256. i)T + (-1.95e4 + 2.25e4i)T^{2} \) |
| 37 | \( 1 + (-114. - 131. i)T + (-7.20e3 + 5.01e4i)T^{2} \) |
| 41 | \( 1 + (-66.9 + 77.2i)T + (-9.80e3 - 6.82e4i)T^{2} \) |
| 43 | \( 1 + (-53.4 + 116. i)T + (-5.20e4 - 6.00e4i)T^{2} \) |
| 47 | \( 1 + 130.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-81.5 + 52.3i)T + (6.18e4 - 1.35e5i)T^{2} \) |
| 59 | \( 1 + (232. + 149. i)T + (8.53e4 + 1.86e5i)T^{2} \) |
| 61 | \( 1 + (-190. - 416. i)T + (-1.48e5 + 1.71e5i)T^{2} \) |
| 67 | \( 1 + (-49.8 - 346. i)T + (-2.88e5 + 8.47e4i)T^{2} \) |
| 71 | \( 1 + (-109. - 758. i)T + (-3.43e5 + 1.00e5i)T^{2} \) |
| 73 | \( 1 + (869. - 255. i)T + (3.27e5 - 2.10e5i)T^{2} \) |
| 79 | \( 1 + (-377. - 242. i)T + (2.04e5 + 4.48e5i)T^{2} \) |
| 83 | \( 1 + (-281. - 325. i)T + (-8.13e4 + 5.65e5i)T^{2} \) |
| 89 | \( 1 + (-138. + 304. i)T + (-4.61e5 - 5.32e5i)T^{2} \) |
| 97 | \( 1 + (557. - 643. i)T + (-1.29e5 - 9.03e5i)T^{2} \) |
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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
−11.53411594869063921726899296351, −10.95402748548368938826862714403, −9.968206694564243433538636325810, −8.815250458628958308879939835678, −7.63598624687169697065128556417, −6.69763868665217013631747182333, −5.39027820906953369967291822340, −4.36660853595352954114891831294, −2.30363031930109167690186342035, −1.20336911060629110762256814992,
1.91877875725409214052666005709, 2.79170225218408386468376018089, 4.70736534474937193189647668324, 6.11930919350321159460891005576, 6.79952613147850209406786522113, 8.139803575903743045544602292083, 8.914016130118742499628511480119, 10.64145599970068944644962053124, 10.93866077968980363419277313251, 11.96519936373915763546881458143
