| L(s) = 1 | + (2.43 + 0.213i)2-s + (3.56 + 3.77i)3-s + (−1.98 − 0.349i)4-s + (−9.02 + 6.59i)5-s + (7.88 + 9.97i)6-s + (16.2 + 11.4i)7-s + (−23.6 − 6.34i)8-s + (−1.54 + 26.9i)9-s + (−23.4 + 14.1i)10-s + (2.74 − 7.54i)11-s + (−5.74 − 8.73i)12-s + (6.05 + 69.1i)13-s + (37.2 + 31.2i)14-s + (−57.1 − 10.5i)15-s + (−41.2 − 14.9i)16-s + (8.95 − 2.40i)17-s + ⋯ |
| L(s) = 1 | + (0.861 + 0.0754i)2-s + (0.686 + 0.727i)3-s + (−0.247 − 0.0436i)4-s + (−0.807 + 0.590i)5-s + (0.536 + 0.678i)6-s + (0.879 + 0.616i)7-s + (−1.04 − 0.280i)8-s + (−0.0574 + 0.998i)9-s + (−0.740 + 0.447i)10-s + (0.0753 − 0.206i)11-s + (−0.138 − 0.210i)12-s + (0.129 + 1.47i)13-s + (0.711 + 0.597i)14-s + (−0.983 − 0.181i)15-s + (−0.643 − 0.234i)16-s + (0.127 − 0.0342i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.306 - 0.951i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.306 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.37492 + 1.88691i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.37492 + 1.88691i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-3.56 - 3.77i)T \) |
| 5 | \( 1 + (9.02 - 6.59i)T \) |
| good | 2 | \( 1 + (-2.43 - 0.213i)T + (7.87 + 1.38i)T^{2} \) |
| 7 | \( 1 + (-16.2 - 11.4i)T + (117. + 322. i)T^{2} \) |
| 11 | \( 1 + (-2.74 + 7.54i)T + (-1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (-6.05 - 69.1i)T + (-2.16e3 + 381. i)T^{2} \) |
| 17 | \( 1 + (-8.95 + 2.40i)T + (4.25e3 - 2.45e3i)T^{2} \) |
| 19 | \( 1 + (1.99 - 1.15i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-2.50 - 3.57i)T + (-4.16e3 + 1.14e4i)T^{2} \) |
| 29 | \( 1 + (-220. + 184. i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-42.4 + 240. i)T + (-2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (-70.6 - 263. i)T + (-4.38e4 + 2.53e4i)T^{2} \) |
| 41 | \( 1 + (-54.8 + 65.3i)T + (-1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (-196. - 422. i)T + (-5.11e4 + 6.09e4i)T^{2} \) |
| 47 | \( 1 + (25.2 - 35.9i)T + (-3.55e4 - 9.75e4i)T^{2} \) |
| 53 | \( 1 + (-94.9 + 94.9i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (-672. + 244. i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-16.1 - 91.6i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (704. - 61.6i)T + (2.96e5 - 5.22e4i)T^{2} \) |
| 71 | \( 1 + (1.01e3 + 583. i)T + (1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-78.8 + 294. i)T + (-3.36e5 - 1.94e5i)T^{2} \) |
| 79 | \( 1 + (-698. - 832. i)T + (-8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (12.3 - 140. i)T + (-5.63e5 - 9.92e4i)T^{2} \) |
| 89 | \( 1 + (292. + 505. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-767. + 357. i)T + (5.86e5 - 6.99e5i)T^{2} \) |
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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
−13.43600580610693205776395818057, −11.87353841243110424338182972882, −11.37274213981850229424020854968, −9.875261361911556137697595574098, −8.801312769797573843799990606786, −7.930675145719071435459266433725, −6.26620112181347070686365034940, −4.71330432666596115520243234577, −4.06338417281448666937662102126, −2.65260816258248243540366297802,
0.904314444192716745663847216751, 3.12421969975767935377580613384, 4.27164693601444787802592615373, 5.42865530016777520515787332836, 7.20147934926225760361128837154, 8.201865095974562498111574455693, 8.861397633059659127730258471692, 10.59636725682782022320966007715, 12.00710647705671051408167627071, 12.55814850170325277582637485002
