| L(s) = 1 | + (0.730 + 1.86i)2-s + (0.592 + 7.90i)3-s + (−2.93 + 2.72i)4-s + (−4.49 + 3.06i)5-s + (−14.2 + 6.87i)6-s + (−16.9 − 7.56i)7-s + (−7.20 − 3.47i)8-s + (−35.3 + 5.33i)9-s + (−8.98 − 6.12i)10-s + (69.1 + 10.4i)11-s + (−23.2 − 21.5i)12-s + (8.79 − 11.0i)13-s + (1.72 − 37.0i)14-s + (−26.8 − 33.6i)15-s + (1.19 − 15.9i)16-s + (−45.2 + 13.9i)17-s + ⋯ |
| L(s) = 1 | + (0.258 + 0.658i)2-s + (0.113 + 1.52i)3-s + (−0.366 + 0.340i)4-s + (−0.401 + 0.273i)5-s + (−0.971 + 0.467i)6-s + (−0.912 − 0.408i)7-s + (−0.318 − 0.153i)8-s + (−1.31 + 0.197i)9-s + (−0.283 − 0.193i)10-s + (1.89 + 0.285i)11-s + (−0.558 − 0.518i)12-s + (0.187 − 0.235i)13-s + (0.0330 − 0.706i)14-s + (−0.462 − 0.579i)15-s + (0.0186 − 0.249i)16-s + (−0.645 + 0.199i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.147914 - 1.29664i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.147914 - 1.29664i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.730 - 1.86i)T \) |
| 7 | \( 1 + (16.9 + 7.56i)T \) |
| good | 3 | \( 1 + (-0.592 - 7.90i)T + (-26.6 + 4.02i)T^{2} \) |
| 5 | \( 1 + (4.49 - 3.06i)T + (45.6 - 116. i)T^{2} \) |
| 11 | \( 1 + (-69.1 - 10.4i)T + (1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (-8.79 + 11.0i)T + (-488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (45.2 - 13.9i)T + (4.05e3 - 2.76e3i)T^{2} \) |
| 19 | \( 1 + (60.5 - 104. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (9.08 + 2.80i)T + (1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (33.1 - 145. i)T + (-2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (94.3 + 163. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-129. - 120. i)T + (3.78e3 + 5.05e4i)T^{2} \) |
| 41 | \( 1 + (80.8 + 38.9i)T + (4.29e4 + 5.38e4i)T^{2} \) |
| 43 | \( 1 + (-392. + 189. i)T + (4.95e4 - 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-19.2 - 48.9i)T + (-7.61e4 + 7.06e4i)T^{2} \) |
| 53 | \( 1 + (-364. + 338. i)T + (1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (-424. - 289. i)T + (7.50e4 + 1.91e5i)T^{2} \) |
| 61 | \( 1 + (-656. - 608. i)T + (1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (126. + 218. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (50.9 + 223. i)T + (-3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (254. - 649. i)T + (-2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (-445. + 772. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (476. + 597. i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (-418. + 63.1i)T + (6.73e5 - 2.07e5i)T^{2} \) |
| 97 | \( 1 - 1.62e3T + 9.12e5T^{2} \) |
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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
−14.56972859567995513959489542581, −13.14405987301381772878783464405, −11.81196692043507911302127806352, −10.58361568922281314506865085207, −9.565137069031699287713781715019, −8.762043054890626960881471254422, −7.06868586712518272777537204724, −5.90092418391459009057942073543, −4.08149869140987586953215975074, −3.72454667315659991621187440342,
0.70194409005485004670551194082, 2.33551465105127276907728022842, 4.03213341773620604277174093656, 6.19226166402507607831706293037, 6.90300462918243630976936897958, 8.581905163031990237580504816234, 9.351354560331361135789791897254, 11.26420313812312540379309036183, 12.00856824670191620925964482643, 12.77149791724014693654355272932
