| L(s) = 1 | + (0.284 + 1.97i)2-s + (−0.493 + 1.08i)3-s + (−3.83 + 1.12i)4-s + (−10.8 + 12.4i)5-s + (−2.27 − 0.669i)6-s + (0.379 − 0.243i)7-s + (−3.32 − 7.27i)8-s + (16.7 + 19.3i)9-s + (−27.8 − 17.8i)10-s + (−1.47 + 10.2i)11-s + (0.676 − 4.70i)12-s + (−24.9 − 16.0i)13-s + (0.590 + 0.681i)14-s + (−8.15 − 17.8i)15-s + (13.4 − 8.65i)16-s + (80.4 + 23.6i)17-s + ⋯ |
| L(s) = 1 | + (0.100 + 0.699i)2-s + (−0.0949 + 0.207i)3-s + (−0.479 + 0.140i)4-s + (−0.968 + 1.11i)5-s + (−0.155 − 0.0455i)6-s + (0.0204 − 0.0131i)7-s + (−0.146 − 0.321i)8-s + (0.620 + 0.716i)9-s + (−0.879 − 0.565i)10-s + (−0.0404 + 0.281i)11-s + (0.0162 − 0.113i)12-s + (−0.533 − 0.342i)13-s + (0.0112 + 0.0130i)14-s + (−0.140 − 0.307i)15-s + (0.210 − 0.135i)16-s + (1.14 + 0.336i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.741 - 0.670i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.741 - 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.378947 + 0.984021i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.378947 + 0.984021i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.284 - 1.97i)T \) |
| 23 | \( 1 + (-61.0 - 91.8i)T \) |
| good | 3 | \( 1 + (0.493 - 1.08i)T + (-17.6 - 20.4i)T^{2} \) |
| 5 | \( 1 + (10.8 - 12.4i)T + (-17.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (-0.379 + 0.243i)T + (142. - 312. i)T^{2} \) |
| 11 | \( 1 + (1.47 - 10.2i)T + (-1.27e3 - 374. i)T^{2} \) |
| 13 | \( 1 + (24.9 + 16.0i)T + (912. + 1.99e3i)T^{2} \) |
| 17 | \( 1 + (-80.4 - 23.6i)T + (4.13e3 + 2.65e3i)T^{2} \) |
| 19 | \( 1 + (-109. + 32.0i)T + (5.77e3 - 3.70e3i)T^{2} \) |
| 29 | \( 1 + (146. + 42.9i)T + (2.05e4 + 1.31e4i)T^{2} \) |
| 31 | \( 1 + (-45.7 - 100. i)T + (-1.95e4 + 2.25e4i)T^{2} \) |
| 37 | \( 1 + (217. + 250. i)T + (-7.20e3 + 5.01e4i)T^{2} \) |
| 41 | \( 1 + (150. - 173. i)T + (-9.80e3 - 6.82e4i)T^{2} \) |
| 43 | \( 1 + (26.5 - 58.0i)T + (-5.20e4 - 6.00e4i)T^{2} \) |
| 47 | \( 1 - 260.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-459. + 295. i)T + (6.18e4 - 1.35e5i)T^{2} \) |
| 59 | \( 1 + (-473. - 304. i)T + (8.53e4 + 1.86e5i)T^{2} \) |
| 61 | \( 1 + (132. + 291. i)T + (-1.48e5 + 1.71e5i)T^{2} \) |
| 67 | \( 1 + (144. + 1.00e3i)T + (-2.88e5 + 8.47e4i)T^{2} \) |
| 71 | \( 1 + (48.3 + 336. i)T + (-3.43e5 + 1.00e5i)T^{2} \) |
| 73 | \( 1 + (-105. + 30.9i)T + (3.27e5 - 2.10e5i)T^{2} \) |
| 79 | \( 1 + (-643. - 413. i)T + (2.04e5 + 4.48e5i)T^{2} \) |
| 83 | \( 1 + (-891. - 1.02e3i)T + (-8.13e4 + 5.65e5i)T^{2} \) |
| 89 | \( 1 + (-324. + 709. i)T + (-4.61e5 - 5.32e5i)T^{2} \) |
| 97 | \( 1 + (121. - 140. i)T + (-1.29e5 - 9.03e5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.55989187322843889126616223012, −14.87980013374220507416975276967, −13.70947545757158021553976158123, −12.22917378054408202405806334464, −10.93007575935902424091443523428, −9.753673845005841765836387595497, −7.73925024173581988073823075171, −7.19015112744902818644642408882, −5.22817515040378587009442612243, −3.49397496742978503102401553726,
0.899035432981467045674695437331, 3.71307080865300741925327875427, 5.17238625719916553757102282962, 7.40955814911650517204277100484, 8.799816342847136939781105950800, 9.990759080231481202282173464384, 11.83287476110065455795638821958, 12.15737797140619229760283461474, 13.34535286343763918161229953573, 14.80364586871842008957646186450
