| L(s) = 1 | + (−0.0173 − 0.0125i)3-s + (−1.51 + 4.65i)5-s + (−1.55 − 2.14i)7-s + (−2.78 − 8.55i)9-s + (−10.0 + 4.45i)11-s + (−7.67 + 2.49i)13-s + (0.0846 − 0.0614i)15-s + (−8.67 − 2.81i)17-s + (17.4 − 24.0i)19-s + 0.0565i·21-s − 28.7·23-s + (0.865 + 0.628i)25-s + (−0.118 + 0.366i)27-s + (−18.0 − 24.8i)29-s + (−13.5 − 41.6i)31-s + ⋯ |
| L(s) = 1 | + (−0.00576 − 0.00418i)3-s + (−0.302 + 0.930i)5-s + (−0.222 − 0.305i)7-s + (−0.309 − 0.951i)9-s + (−0.914 + 0.405i)11-s + (−0.590 + 0.191i)13-s + (0.00564 − 0.00409i)15-s + (−0.510 − 0.165i)17-s + (0.920 − 1.26i)19-s + 0.00269i·21-s − 1.24·23-s + (0.0346 + 0.0251i)25-s + (−0.00440 + 0.0135i)27-s + (−0.622 − 0.856i)29-s + (−0.436 − 1.34i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 + 0.425i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.905 + 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0382533 - 0.171400i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0382533 - 0.171400i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (1.55 + 2.14i)T \) |
| 11 | \( 1 + (10.0 - 4.45i)T \) |
| good | 3 | \( 1 + (0.0173 + 0.0125i)T + (2.78 + 8.55i)T^{2} \) |
| 5 | \( 1 + (1.51 - 4.65i)T + (-20.2 - 14.6i)T^{2} \) |
| 13 | \( 1 + (7.67 - 2.49i)T + (136. - 99.3i)T^{2} \) |
| 17 | \( 1 + (8.67 + 2.81i)T + (233. + 169. i)T^{2} \) |
| 19 | \( 1 + (-17.4 + 24.0i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + 28.7T + 529T^{2} \) |
| 29 | \( 1 + (18.0 + 24.8i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (13.5 + 41.6i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (43.3 - 31.4i)T + (423. - 1.30e3i)T^{2} \) |
| 41 | \( 1 + (35.0 - 48.3i)T + (-519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 - 58.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (14.9 + 10.8i)T + (682. + 2.10e3i)T^{2} \) |
| 53 | \( 1 + (11.5 + 35.5i)T + (-2.27e3 + 1.65e3i)T^{2} \) |
| 59 | \( 1 + (-20.7 + 15.0i)T + (1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-17.8 - 5.79i)T + (3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 - 104.T + 4.48e3T^{2} \) |
| 71 | \( 1 + (23.4 - 72.0i)T + (-4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (53.2 + 73.3i)T + (-1.64e3 + 5.06e3i)T^{2} \) |
| 79 | \( 1 + (-110. + 35.7i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-123. - 40.2i)T + (5.57e3 + 4.04e3i)T^{2} \) |
| 89 | \( 1 + 94.4T + 7.92e3T^{2} \) |
| 97 | \( 1 + (1.26 + 3.88i)T + (-7.61e3 + 5.53e3i)T^{2} \) |
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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
−11.25762692675901947441443011629, −10.02859277538819433045163264741, −9.458704866721817871050659847954, −8.022317058060246940156725866702, −7.13238419963836082963640550074, −6.35048789300694801716152914553, −4.94658622223121869620279736809, −3.58217535254838719912184531908, −2.50924430726634767963380728123, −0.07831390311827011561691172154,
2.01090685447703728470020032392, 3.56903254378200830602004641905, 5.10571507679106912166347591595, 5.56151659619408982909777535086, 7.26510569000791585474441065791, 8.196241728756024576799608342914, 8.838661578077199834909432780530, 10.13065802712420094408972346498, 10.84008960562321085579558645156, 12.16784192561364129200381139212
