| L(s) = 1 | + (0.347 − 1.96i)2-s + (2.29 + 1.92i)3-s + (−3.75 − 1.36i)4-s + (3.13 − 1.14i)5-s + (4.59 − 3.85i)6-s + (−16.8 − 29.1i)7-s + (−4 + 6.92i)8-s + (1.56 + 8.86i)9-s + (−1.15 − 6.57i)10-s + (12.2 − 21.1i)11-s + (−6 − 10.3i)12-s + (47.4 − 39.8i)13-s + (−63.2 + 23.0i)14-s + (9.40 + 3.42i)15-s + (12.2 + 10.2i)16-s + (20.2 − 114. i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (0.442 + 0.371i)3-s + (−0.469 − 0.171i)4-s + (0.280 − 0.102i)5-s + (0.312 − 0.262i)6-s + (−0.908 − 1.57i)7-s + (−0.176 + 0.306i)8-s + (0.0578 + 0.328i)9-s + (−0.0366 − 0.207i)10-s + (0.335 − 0.580i)11-s + (−0.144 − 0.249i)12-s + (1.01 − 0.849i)13-s + (−1.20 + 0.439i)14-s + (0.161 + 0.0589i)15-s + (0.191 + 0.160i)16-s + (0.288 − 1.63i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.962475 - 1.37501i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.962475 - 1.37501i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.347 + 1.96i)T \) |
| 3 | \( 1 + (-2.29 - 1.92i)T \) |
| 19 | \( 1 + (-1.28 - 82.8i)T \) |
| good | 5 | \( 1 + (-3.13 + 1.14i)T + (95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (16.8 + 29.1i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-12.2 + 21.1i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-47.4 + 39.8i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-20.2 + 114. i)T + (-4.61e3 - 1.68e3i)T^{2} \) |
| 23 | \( 1 + (104. + 38.1i)T + (9.32e3 + 7.82e3i)T^{2} \) |
| 29 | \( 1 + (-18.8 - 107. i)T + (-2.29e4 + 8.34e3i)T^{2} \) |
| 31 | \( 1 + (-72.5 - 125. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 134.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-54.5 - 45.7i)T + (1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + (-232. + 84.5i)T + (6.09e4 - 5.11e4i)T^{2} \) |
| 47 | \( 1 + (73.5 + 417. i)T + (-9.75e4 + 3.55e4i)T^{2} \) |
| 53 | \( 1 + (-547. - 199. i)T + (1.14e5 + 9.56e4i)T^{2} \) |
| 59 | \( 1 + (28.6 - 162. i)T + (-1.92e5 - 7.02e4i)T^{2} \) |
| 61 | \( 1 + (-650. - 236. i)T + (1.73e5 + 1.45e5i)T^{2} \) |
| 67 | \( 1 + (-179. - 1.01e3i)T + (-2.82e5 + 1.02e5i)T^{2} \) |
| 71 | \( 1 + (509. - 185. i)T + (2.74e5 - 2.30e5i)T^{2} \) |
| 73 | \( 1 + (511. + 429. i)T + (6.75e4 + 3.83e5i)T^{2} \) |
| 79 | \( 1 + (180. + 151. i)T + (8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (320. + 554. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-840. + 705. i)T + (1.22e5 - 6.94e5i)T^{2} \) |
| 97 | \( 1 + (144. - 821. i)T + (-8.57e5 - 3.12e5i)T^{2} \) |
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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
−13.07053010535984366895849625087, −11.65399633707325160781412903497, −10.39762625565406144924189675180, −9.966219812451737430541344814641, −8.710644074670936059062002307855, −7.34643638511031632042450513110, −5.78721069911158041776730328257, −4.04006666494112969779423197394, −3.18390221172902736230423299725, −0.877245660074129718237863029423,
2.22718965673757040396347731520, 3.96223011050627359986193817609, 5.98423507164346667669806484261, 6.41128737549228752316358939141, 8.062575360414279826421500834586, 9.019275493295324511314868308982, 9.793989317492436974740735093323, 11.67587808890239618614709353065, 12.63758172499407833497920649345, 13.42878678180843215899403705537
