| L(s) = 1 | + (1.53 + 1.28i)2-s + (2.81 − 1.02i)3-s + (0.694 + 3.93i)4-s + (3.09 − 17.5i)5-s + (5.63 + 2.05i)6-s + (7.98 + 13.8i)7-s + (−4.00 + 6.92i)8-s + (6.89 − 5.78i)9-s + (27.3 − 22.9i)10-s + (24.7 − 42.9i)11-s + (6 + 10.3i)12-s + (15.0 + 5.48i)13-s + (−5.54 + 31.4i)14-s + (−9.29 − 52.6i)15-s + (−15.0 + 5.47i)16-s + (−28.4 − 23.8i)17-s + ⋯ |
| L(s) = 1 | + (0.541 + 0.454i)2-s + (0.542 − 0.197i)3-s + (0.0868 + 0.492i)4-s + (0.277 − 1.57i)5-s + (0.383 + 0.139i)6-s + (0.430 + 0.746i)7-s + (−0.176 + 0.306i)8-s + (0.255 − 0.214i)9-s + (0.864 − 0.725i)10-s + (0.679 − 1.17i)11-s + (0.144 + 0.249i)12-s + (0.321 + 0.117i)13-s + (−0.105 + 0.600i)14-s + (−0.159 − 0.907i)15-s + (−0.234 + 0.0855i)16-s + (−0.405 − 0.340i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.71258 - 0.164277i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.71258 - 0.164277i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.53 - 1.28i)T \) |
| 3 | \( 1 + (-2.81 + 1.02i)T \) |
| 19 | \( 1 + (-31.2 - 76.6i)T \) |
| good | 5 | \( 1 + (-3.09 + 17.5i)T + (-117. - 42.7i)T^{2} \) |
| 7 | \( 1 + (-7.98 - 13.8i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-24.7 + 42.9i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-15.0 - 5.48i)T + (1.68e3 + 1.41e3i)T^{2} \) |
| 17 | \( 1 + (28.4 + 23.8i)T + (853. + 4.83e3i)T^{2} \) |
| 23 | \( 1 + (-31.7 - 179. i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (34.8 - 29.2i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (90.7 + 157. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 294.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (77.9 - 28.3i)T + (5.27e4 - 4.43e4i)T^{2} \) |
| 43 | \( 1 + (58.7 - 333. i)T + (-7.47e4 - 2.71e4i)T^{2} \) |
| 47 | \( 1 + (-26.0 + 21.8i)T + (1.80e4 - 1.02e5i)T^{2} \) |
| 53 | \( 1 + (-30.3 - 172. i)T + (-1.39e5 + 5.09e4i)T^{2} \) |
| 59 | \( 1 + (669. + 562. i)T + (3.56e4 + 2.02e5i)T^{2} \) |
| 61 | \( 1 + (-83.9 - 476. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (419. - 351. i)T + (5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-67.6 + 383. i)T + (-3.36e5 - 1.22e5i)T^{2} \) |
| 73 | \( 1 + (556. - 202. i)T + (2.98e5 - 2.50e5i)T^{2} \) |
| 79 | \( 1 + (-916. + 333. i)T + (3.77e5 - 3.16e5i)T^{2} \) |
| 83 | \( 1 + (-643. - 1.11e3i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-1.27e3 - 464. i)T + (5.40e5 + 4.53e5i)T^{2} \) |
| 97 | \( 1 + (14.7 + 12.3i)T + (1.58e5 + 8.98e5i)T^{2} \) |
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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
−13.33402076061323228292082831008, −12.23895755771386056697097383766, −11.46371873056703915411304215366, −9.314831517600404824688329170146, −8.754464560123521420241623535337, −7.80186475054831132392541574640, −6.01686464702033581062326253951, −5.12803793477551204875246780764, −3.63364585237739453492623255973, −1.52787126122991743885811439311,
2.04343664035877812132078927271, 3.40201457437893630366462809974, 4.63121031362644639389888425822, 6.59059642433313741133051502829, 7.28767850905323005915333160772, 9.066956484960822703942417241918, 10.44663231277923397623244121300, 10.70760743417199679253929208975, 12.06210119096625636777030723024, 13.39131713764411509532381839020
