| L(s) = 1 | + (1.68 + 1.08i)2-s + (0.0135 + 0.0940i)3-s + (1.66 + 3.63i)4-s + (11.1 + 3.27i)5-s + (−0.0789 + 0.172i)6-s + (−6.36 + 7.35i)7-s + (−1.13 + 7.91i)8-s + (25.8 − 7.60i)9-s + (15.2 + 17.5i)10-s + (−15.8 + 10.2i)11-s + (−0.319 + 0.205i)12-s + (−52.0 − 60.1i)13-s + (−18.6 + 5.48i)14-s + (−0.157 + 1.09i)15-s + (−10.4 + 12.0i)16-s + (12.6 − 27.6i)17-s + ⋯ |
| L(s) = 1 | + (0.594 + 0.382i)2-s + (0.00260 + 0.0180i)3-s + (0.207 + 0.454i)4-s + (0.998 + 0.293i)5-s + (−0.00537 + 0.0117i)6-s + (−0.343 + 0.396i)7-s + (−0.0503 + 0.349i)8-s + (0.959 − 0.281i)9-s + (0.481 + 0.556i)10-s + (−0.435 + 0.279i)11-s + (−0.00769 + 0.00494i)12-s + (−1.11 − 1.28i)13-s + (−0.356 + 0.104i)14-s + (−0.00270 + 0.0188i)15-s + (−0.163 + 0.188i)16-s + (0.180 − 0.395i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 - 0.675i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.82244 + 0.708311i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.82244 + 0.708311i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.68 - 1.08i)T \) |
| 23 | \( 1 + (29.6 + 106. i)T \) |
| good | 3 | \( 1 + (-0.0135 - 0.0940i)T + (-25.9 + 7.60i)T^{2} \) |
| 5 | \( 1 + (-11.1 - 3.27i)T + (105. + 67.5i)T^{2} \) |
| 7 | \( 1 + (6.36 - 7.35i)T + (-48.8 - 339. i)T^{2} \) |
| 11 | \( 1 + (15.8 - 10.2i)T + (552. - 1.21e3i)T^{2} \) |
| 13 | \( 1 + (52.0 + 60.1i)T + (-312. + 2.17e3i)T^{2} \) |
| 17 | \( 1 + (-12.6 + 27.6i)T + (-3.21e3 - 3.71e3i)T^{2} \) |
| 19 | \( 1 + (12.8 + 28.2i)T + (-4.49e3 + 5.18e3i)T^{2} \) |
| 29 | \( 1 + (37.6 - 82.5i)T + (-1.59e4 - 1.84e4i)T^{2} \) |
| 31 | \( 1 + (28.4 - 197. i)T + (-2.85e4 - 8.39e3i)T^{2} \) |
| 37 | \( 1 + (-319. + 93.7i)T + (4.26e4 - 2.73e4i)T^{2} \) |
| 41 | \( 1 + (8.20 + 2.40i)T + (5.79e4 + 3.72e4i)T^{2} \) |
| 43 | \( 1 + (-59.5 - 414. i)T + (-7.62e4 + 2.23e4i)T^{2} \) |
| 47 | \( 1 + 8.49T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-27.3 + 31.5i)T + (-2.11e4 - 1.47e5i)T^{2} \) |
| 59 | \( 1 + (79.6 + 91.9i)T + (-2.92e4 + 2.03e5i)T^{2} \) |
| 61 | \( 1 + (-53.7 + 373. i)T + (-2.17e5 - 6.39e4i)T^{2} \) |
| 67 | \( 1 + (333. + 214. i)T + (1.24e5 + 2.73e5i)T^{2} \) |
| 71 | \( 1 + (947. + 608. i)T + (1.48e5 + 3.25e5i)T^{2} \) |
| 73 | \( 1 + (9.05 + 19.8i)T + (-2.54e5 + 2.93e5i)T^{2} \) |
| 79 | \( 1 + (-865. - 999. i)T + (-7.01e4 + 4.88e5i)T^{2} \) |
| 83 | \( 1 + (539. - 158. i)T + (4.81e5 - 3.09e5i)T^{2} \) |
| 89 | \( 1 + (-49.7 - 346. i)T + (-6.76e5 + 1.98e5i)T^{2} \) |
| 97 | \( 1 + (-1.35e3 - 397. i)T + (7.67e5 + 4.93e5i)T^{2} \) |
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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
−15.27050171143486715854763613301, −14.37355332466145950442515425229, −13.00313781305341982387865302404, −12.44953488124930777730594354866, −10.48090237177341943513381810883, −9.510053250644504633413701365278, −7.59707488454281857379689109225, −6.28133050141035203798222344171, −4.93637229469492114448089340645, −2.69084010979947419654948700719,
1.96882008837639800866160060189, 4.27479473664763436814890793751, 5.81139381199628919321786629057, 7.32746831916880317250508451640, 9.513429524584530117443864835024, 10.21487407069092028096984835118, 11.79169523380396435241175360475, 13.10716970663508200513552959601, 13.64974865950406565637441536563, 14.92410730719894218830706033003
