| L(s) = 1 | + (−1.80 − 0.867i)2-s + (−0.791 + 0.992i)3-s + (2.49 + 3.12i)4-s + (−2.81 − 1.35i)5-s + (2.28 − 1.10i)6-s + (−12.3 + 15.4i)7-s + (−1.78 − 7.79i)8-s + (5.64 + 24.7i)9-s + (3.89 + 4.88i)10-s + (−4.23 + 18.5i)11-s − 5.07·12-s + (−14.6 + 64.3i)13-s + (35.6 − 17.1i)14-s + (3.57 − 1.72i)15-s + (−3.56 + 15.5i)16-s + 37.9·17-s + ⋯ |
| L(s) = 1 | + (−0.637 − 0.306i)2-s + (−0.152 + 0.190i)3-s + (0.311 + 0.390i)4-s + (−0.252 − 0.121i)5-s + (0.155 − 0.0749i)6-s + (−0.665 + 0.834i)7-s + (−0.0786 − 0.344i)8-s + (0.209 + 0.916i)9-s + (0.123 + 0.154i)10-s + (−0.116 + 0.508i)11-s − 0.122·12-s + (−0.313 + 1.37i)13-s + (0.680 − 0.327i)14-s + (0.0615 − 0.0296i)15-s + (−0.0556 + 0.243i)16-s + 0.541·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.143 - 0.989i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.143 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.432335 + 0.499657i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.432335 + 0.499657i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.80 + 0.867i)T \) |
| 29 | \( 1 + (-117. + 103. i)T \) |
| good | 3 | \( 1 + (0.791 - 0.992i)T + (-6.00 - 26.3i)T^{2} \) |
| 5 | \( 1 + (2.81 + 1.35i)T + (77.9 + 97.7i)T^{2} \) |
| 7 | \( 1 + (12.3 - 15.4i)T + (-76.3 - 334. i)T^{2} \) |
| 11 | \( 1 + (4.23 - 18.5i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (14.6 - 64.3i)T + (-1.97e3 - 953. i)T^{2} \) |
| 17 | \( 1 - 37.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + (52.2 + 65.5i)T + (-1.52e3 + 6.68e3i)T^{2} \) |
| 23 | \( 1 + (30.8 - 14.8i)T + (7.58e3 - 9.51e3i)T^{2} \) |
| 31 | \( 1 + (-105. - 50.9i)T + (1.85e4 + 2.32e4i)T^{2} \) |
| 37 | \( 1 + (-33.3 - 146. i)T + (-4.56e4 + 2.19e4i)T^{2} \) |
| 41 | \( 1 - 272.T + 6.89e4T^{2} \) |
| 43 | \( 1 + (357. - 172. i)T + (4.95e4 - 6.21e4i)T^{2} \) |
| 47 | \( 1 + (24.6 - 108. i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (-440. - 212. i)T + (9.28e4 + 1.16e5i)T^{2} \) |
| 59 | \( 1 + 301.T + 2.05e5T^{2} \) |
| 61 | \( 1 + (14.8 - 18.5i)T + (-5.05e4 - 2.21e5i)T^{2} \) |
| 67 | \( 1 + (1.73 + 7.61i)T + (-2.70e5 + 1.30e5i)T^{2} \) |
| 71 | \( 1 + (-169. + 742. i)T + (-3.22e5 - 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-640. + 308. i)T + (2.42e5 - 3.04e5i)T^{2} \) |
| 79 | \( 1 + (-205. - 901. i)T + (-4.44e5 + 2.13e5i)T^{2} \) |
| 83 | \( 1 + (-3.83 - 4.80i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (-681. - 328. i)T + (4.39e5 + 5.51e5i)T^{2} \) |
| 97 | \( 1 + (189. + 237. i)T + (-2.03e5 + 8.89e5i)T^{2} \) |
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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
−15.32857520506665537612047651366, −13.74167228625941199447291804415, −12.41799387093619071554149588608, −11.57909704570430116656281702215, −10.20177221036495606485750496623, −9.253749223550532792469922985509, −7.953687293782366390043651264950, −6.47423043836176265796338782167, −4.55690858007792111514587597596, −2.35394817679786528102721862944,
0.57998487950993517569804200003, 3.51631242218531294632910238102, 5.85533250685766709706599246759, 7.08454869756972525976510970846, 8.223833430896359000697718542309, 9.793713679639322281349941618030, 10.59718844268656596492186840049, 12.09507287885958942146262254357, 13.18022752132255764297894128445, 14.60316664761948462839768806048
