| L(s) = 1 | + (−0.415 + 0.909i)2-s + (0.959 + 0.281i)3-s + (−0.654 − 0.755i)4-s + (0.841 − 0.540i)5-s + (−0.654 + 0.755i)6-s + (0.223 − 1.55i)7-s + (0.959 − 0.281i)8-s + (0.841 + 0.540i)9-s + (0.142 + 0.989i)10-s + (−1.52 − 3.34i)11-s + (−0.415 − 0.909i)12-s + (−0.599 − 4.16i)13-s + (1.32 + 0.848i)14-s + (0.959 − 0.281i)15-s + (−0.142 + 0.989i)16-s + (2.64 − 3.04i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 + 0.643i)2-s + (0.553 + 0.162i)3-s + (−0.327 − 0.377i)4-s + (0.376 − 0.241i)5-s + (−0.267 + 0.308i)6-s + (0.0844 − 0.587i)7-s + (0.339 − 0.0996i)8-s + (0.280 + 0.180i)9-s + (0.0450 + 0.313i)10-s + (−0.460 − 1.00i)11-s + (−0.119 − 0.262i)12-s + (−0.166 − 1.15i)13-s + (0.353 + 0.226i)14-s + (0.247 − 0.0727i)15-s + (−0.0355 + 0.247i)16-s + (0.640 − 0.739i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.45019 - 0.334103i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.45019 - 0.334103i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.415 - 0.909i)T \) |
| 3 | \( 1 + (-0.959 - 0.281i)T \) |
| 5 | \( 1 + (-0.841 + 0.540i)T \) |
| 23 | \( 1 + (0.237 - 4.78i)T \) |
| good | 7 | \( 1 + (-0.223 + 1.55i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (1.52 + 3.34i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (0.599 + 4.16i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-2.64 + 3.04i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (2.25 + 2.59i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (1.39 - 1.61i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-1.47 + 0.432i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (4.85 + 3.11i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (-6.62 + 4.25i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (5.85 + 1.71i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 + (1.02 - 7.15i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-1.86 - 12.9i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-7.12 + 2.09i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (3.14 - 6.88i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-3.98 + 8.71i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (3.41 + 3.94i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-1.31 - 9.12i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-2.22 - 1.42i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-4.50 - 1.32i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (9.89 - 6.35i)T + (40.2 - 88.2i)T^{2} \) |
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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
−10.36928557870145679635020987283, −9.352260824956889192864018335640, −8.677176084700488014926394456608, −7.75746099814975377019962663218, −7.18784994976617566292155912098, −5.79228064322822347584538900036, −5.20625013873089994963512921355, −3.87071579934634993891305702771, −2.69814100557286253988224676545, −0.840783555814318140413181714410,
1.79807833465041280038926586155, 2.46826073486117951847401187292, 3.82340318111811607541502629074, 4.85049097591437203438503584918, 6.17716410635488661369006848188, 7.15895159067930886380222197541, 8.169030030376352199829723551530, 8.861293650308673156267003873530, 9.822175787915464033779406703706, 10.26084584667414211543423248022
