| L(s) = 1 | + (−4.50 + 13.8i)3-s + 78.4·5-s + (−14.7 + 10.7i)7-s + (24.3 + 17.6i)9-s + (497. − 361. i)11-s + (79.1 − 243. i)13-s + (−353. + 1.08e3i)15-s + (438. + 318. i)17-s + (−849. − 2.61e3i)19-s + (−82.4 − 253. i)21-s + (1.68e3 + 1.22e3i)23-s + 3.02e3·25-s + (−3.22e3 + 2.34e3i)27-s + (2.58e3 + 7.95e3i)29-s + (1.02e3 + 5.25e3i)31-s + ⋯ |
| L(s) = 1 | + (−0.289 + 0.890i)3-s + 1.40·5-s + (−0.114 + 0.0828i)7-s + (0.100 + 0.0727i)9-s + (1.24 − 0.901i)11-s + (0.129 − 0.399i)13-s + (−0.405 + 1.24i)15-s + (0.368 + 0.267i)17-s + (−0.540 − 1.66i)19-s + (−0.0407 − 0.125i)21-s + (0.662 + 0.481i)23-s + 0.968·25-s + (−0.851 + 0.618i)27-s + (0.570 + 1.75i)29-s + (0.192 + 0.981i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.708 - 0.705i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.708 - 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.436045724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.436045724\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 + (-1.02e3 - 5.25e3i)T \) |
| good | 3 | \( 1 + (4.50 - 13.8i)T + (-196. - 142. i)T^{2} \) |
| 5 | \( 1 - 78.4T + 3.12e3T^{2} \) |
| 7 | \( 1 + (14.7 - 10.7i)T + (5.19e3 - 1.59e4i)T^{2} \) |
| 11 | \( 1 + (-497. + 361. i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-79.1 + 243. i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-438. - 318. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (849. + 2.61e3i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-1.68e3 - 1.22e3i)T + (1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-2.58e3 - 7.95e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 37 | \( 1 - 1.49e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-1.22e3 - 3.76e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 + (6.50e3 + 2.00e4i)T + (-1.18e8 + 8.64e7i)T^{2} \) |
| 47 | \( 1 + (1.53e3 - 4.71e3i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (7.98e3 + 5.80e3i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (1.76e3 - 5.43e3i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + 3.93e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.11e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-6.29e4 - 4.57e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-1.52e4 + 1.10e4i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (7.15e4 + 5.19e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (7.91e3 + 2.43e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + (2.25e4 - 1.64e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-1.82e3 + 1.32e3i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−12.76771338731279898997356818343, −11.22748796661969713463957177470, −10.51515587216548134621855440933, −9.453440108213935090618361163098, −8.822307513739664238059381288792, −6.81309769293075665868526460147, −5.77996336238176231535669592451, −4.72242472914607566498620194096, −3.12165021708400203593969240823, −1.29392505368495758361582157207,
1.17887329233378121182022522449, 2.12657209521632411632251089592, 4.27907530935176038797725650102, 6.10826579624536067834291084375, 6.47766189522291529681373937481, 7.85338942692511960648900671604, 9.507458187647334476746252298219, 9.932032786827357716459008453013, 11.57586141567443353527978330246, 12.47213177911082485451498633857
