| L(s) = 1 | + (−1.33 − 2.30i)2-s + (0.537 − 5.16i)3-s + (0.454 − 0.787i)4-s + 19.2·5-s + (−12.6 + 5.64i)6-s + (18.0 − 4.02i)7-s − 23.7·8-s + (−26.4 − 5.55i)9-s + (−25.6 − 44.3i)10-s − 38.5·11-s + (−3.82 − 2.77i)12-s + (38.2 + 66.3i)13-s + (−33.3 − 36.3i)14-s + (10.3 − 99.4i)15-s + (27.9 + 48.4i)16-s + (−11.9 − 20.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.470 − 0.815i)2-s + (0.103 − 0.994i)3-s + (0.0568 − 0.0983i)4-s + 1.72·5-s + (−0.859 + 0.383i)6-s + (0.976 − 0.217i)7-s − 1.04·8-s + (−0.978 − 0.205i)9-s + (−0.810 − 1.40i)10-s − 1.05·11-s + (−0.0919 − 0.0666i)12-s + (0.816 + 1.41i)13-s + (−0.636 − 0.693i)14-s + (0.178 − 1.71i)15-s + (0.436 + 0.756i)16-s + (−0.170 − 0.294i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.583 + 0.811i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.583 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.696475 - 1.35840i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.696475 - 1.35840i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.537 + 5.16i)T \) |
| 7 | \( 1 + (-18.0 + 4.02i)T \) |
| good | 2 | \( 1 + (1.33 + 2.30i)T + (-4 + 6.92i)T^{2} \) |
| 5 | \( 1 - 19.2T + 125T^{2} \) |
| 11 | \( 1 + 38.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + (-38.2 - 66.3i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (11.9 + 20.6i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (23.6 - 41.0i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + 13.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + (53.5 - 92.6i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-79.3 + 137. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-23.0 + 39.8i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-101. - 175. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (42.1 - 72.9i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (236. + 410. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (39.8 + 69.0i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-158. + 274. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-81.8 - 141. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (270. - 467. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 810.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (142. + 246. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (367. + 635. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (290. - 503. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (463. - 803. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (413. - 715. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−13.86114257142942996092374959620, −13.04130189271651886094265435114, −11.61891136459377321833752625181, −10.72623539353959444592446458878, −9.529742981930146919351981428339, −8.422297837592870027159654528536, −6.61670041383751173128662571741, −5.52711756318717106041746615358, −2.31627856685167987067369783773, −1.48317083427460745636174173350,
2.69127042090109608461447388650, 5.23871886480392930746770781180, 6.02972130447035964809398669105, 8.032722863359297405888978845599, 8.883944531545785328252780930280, 10.12288899142522825209463394129, 11.00726338765178904910392179750, 12.89306018650951280378500800075, 14.04147456269357930295768875747, 15.15674509790890568067127223820
