| L(s) = 1 | + (1.46 − 1.36i)2-s + (6.59 − 0.994i)3-s + (0.298 − 3.98i)4-s + (1.69 − 4.31i)5-s + (8.31 − 10.4i)6-s + (−10.9 − 14.9i)7-s + (−4.98 − 6.25i)8-s + (16.7 − 5.15i)9-s + (−3.38 − 8.63i)10-s + (24.4 + 7.55i)11-s + (−1.99 − 26.6i)12-s + (12.5 + 55.1i)13-s + (−36.3 − 6.98i)14-s + (6.88 − 30.1i)15-s + (−15.8 − 2.38i)16-s + (12.7 − 8.71i)17-s + ⋯ |
| L(s) = 1 | + (0.518 − 0.480i)2-s + (1.26 − 0.191i)3-s + (0.0373 − 0.498i)4-s + (0.151 − 0.386i)5-s + (0.565 − 0.709i)6-s + (−0.591 − 0.806i)7-s + (−0.220 − 0.276i)8-s + (0.619 − 0.191i)9-s + (−0.107 − 0.272i)10-s + (0.671 + 0.207i)11-s + (−0.0479 − 0.640i)12-s + (0.268 + 1.17i)13-s + (−0.694 − 0.133i)14-s + (0.118 − 0.519i)15-s + (−0.247 − 0.0372i)16-s + (0.182 − 0.124i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.330 + 0.943i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.330 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.32248 - 1.64794i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.32248 - 1.64794i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.46 + 1.36i)T \) |
| 7 | \( 1 + (10.9 + 14.9i)T \) |
| good | 3 | \( 1 + (-6.59 + 0.994i)T + (25.8 - 7.95i)T^{2} \) |
| 5 | \( 1 + (-1.69 + 4.31i)T + (-91.6 - 85.0i)T^{2} \) |
| 11 | \( 1 + (-24.4 - 7.55i)T + (1.09e3 + 749. i)T^{2} \) |
| 13 | \( 1 + (-12.5 - 55.1i)T + (-1.97e3 + 953. i)T^{2} \) |
| 17 | \( 1 + (-12.7 + 8.71i)T + (1.79e3 - 4.57e3i)T^{2} \) |
| 19 | \( 1 + (-27.7 - 48.0i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (35.8 + 24.4i)T + (4.44e3 + 1.13e4i)T^{2} \) |
| 29 | \( 1 + (48.7 + 23.4i)T + (1.52e4 + 1.90e4i)T^{2} \) |
| 31 | \( 1 + (-45.1 + 78.1i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (11.5 + 154. i)T + (-5.00e4 + 7.54e3i)T^{2} \) |
| 41 | \( 1 + (-140. - 176. i)T + (-1.53e4 + 6.71e4i)T^{2} \) |
| 43 | \( 1 + (250. - 313. i)T + (-1.76e4 - 7.75e4i)T^{2} \) |
| 47 | \( 1 + (80.5 - 74.7i)T + (7.75e3 - 1.03e5i)T^{2} \) |
| 53 | \( 1 + (50.4 - 672. i)T + (-1.47e5 - 2.21e4i)T^{2} \) |
| 59 | \( 1 + (125. + 318. i)T + (-1.50e5 + 1.39e5i)T^{2} \) |
| 61 | \( 1 + (55.7 + 743. i)T + (-2.24e5 + 3.38e4i)T^{2} \) |
| 67 | \( 1 + (8.65 - 14.9i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-555. + 267. i)T + (2.23e5 - 2.79e5i)T^{2} \) |
| 73 | \( 1 + (821. + 762. i)T + (2.90e4 + 3.87e5i)T^{2} \) |
| 79 | \( 1 + (31.1 + 53.9i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-163. + 717. i)T + (-5.15e5 - 2.48e5i)T^{2} \) |
| 89 | \( 1 + (-223. + 69.0i)T + (5.82e5 - 3.97e5i)T^{2} \) |
| 97 | \( 1 - 519.T + 9.12e5T^{2} \) |
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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
−13.41555028321168552350021761551, −12.45890017082441155993826275548, −11.20403644000710833205870553943, −9.719518616098809713126345035223, −9.098534249808131417300389248778, −7.65256938257090305008978813719, −6.35337169490168383963174964518, −4.36222880742099665155826945579, −3.27703397260926393837042853332, −1.59679940296359685231928011118,
2.69577529757958423513253097759, 3.60644468866346828765873412724, 5.53450026258121773043405715473, 6.80030335145221955592141620041, 8.221142906746172486736230539625, 8.992341382917779669164531529020, 10.16610076150277466483093747903, 11.78553727777908025925827529060, 12.95110065753513212885716459046, 13.80568150433756122670182563728
