| L(s) = 1 | + 0.512i·3-s + 2.30·5-s − 4.59·7-s + 8.73·9-s + 13.7·11-s − 22.2i·13-s + 1.18i·15-s + 5.43·17-s + (−7.35 − 17.5i)19-s − 2.35i·21-s − 24.9·23-s − 19.6·25-s + 9.08i·27-s − 34.7i·29-s + 41.2i·31-s + ⋯ |
| L(s) = 1 | + 0.170i·3-s + 0.461·5-s − 0.656·7-s + 0.970·9-s + 1.25·11-s − 1.71i·13-s + 0.0787i·15-s + 0.319·17-s + (−0.386 − 0.922i)19-s − 0.112i·21-s − 1.08·23-s − 0.787·25-s + 0.336i·27-s − 1.19i·29-s + 1.33i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.966705173\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.966705173\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (7.35 + 17.5i)T \) |
| good | 3 | \( 1 - 0.512iT - 9T^{2} \) |
| 5 | \( 1 - 2.30T + 25T^{2} \) |
| 7 | \( 1 + 4.59T + 49T^{2} \) |
| 11 | \( 1 - 13.7T + 121T^{2} \) |
| 13 | \( 1 + 22.2iT - 169T^{2} \) |
| 17 | \( 1 - 5.43T + 289T^{2} \) |
| 23 | \( 1 + 24.9T + 529T^{2} \) |
| 29 | \( 1 + 34.7iT - 841T^{2} \) |
| 31 | \( 1 - 41.2iT - 961T^{2} \) |
| 37 | \( 1 - 20.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 27.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 10.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 33.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 29.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 65.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 107.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 1.78iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 95.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 65.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 37.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 32.6T + 6.88e3T^{2} \) |
| 89 | \( 1 + 74.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 128. iT - 9.40e3T^{2} \) |
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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
−9.713199198950250486365321403983, −8.638037068518774633949248641812, −7.75052644460553944421300554826, −6.76803411352361699390598491939, −6.13541137807833174806714281114, −5.17261924397379157183190704624, −4.06108215161524681471538085824, −3.26128456215709501387821183768, −1.93405965613566348099494904324, −0.60293890468208778625158330266,
1.35977156481053983307067268909, 2.14495846875237428618472502700, 3.89641613729809234950252641655, 4.15254211448741031494740300683, 5.70613354657567982014293104174, 6.49469549623976549398027166340, 6.97569956706894075811148459420, 8.037245693314720770754012980232, 9.161625679164179691489186383056, 9.622055185357631456961229786513
