L(s) = 1 | + 5.19i·3-s − 31.1i·7-s − 27·9-s − 70·13-s − 155. i·19-s + 162·21-s + 125·25-s − 140. i·27-s − 155. i·31-s − 110·37-s − 363. i·39-s + 218. i·43-s − 629·49-s + 810·57-s − 182·61-s + ⋯ |
L(s) = 1 | + 0.999i·3-s − 1.68i·7-s − 9-s − 1.49·13-s − 1.88i·19-s + 1.68·21-s + 25-s − 1.00i·27-s − 0.903i·31-s − 0.488·37-s − 1.49i·39-s + 0.773i·43-s − 1.83·49-s + 1.88·57-s − 0.382·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.649630 - 0.649630i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.649630 - 0.649630i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 5.19iT \) |
good | 5 | \( 1 - 125T^{2} \) |
| 7 | \( 1 + 31.1iT - 343T^{2} \) |
| 11 | \( 1 + 1.33e3T^{2} \) |
| 13 | \( 1 + 70T + 2.19e3T^{2} \) |
| 17 | \( 1 - 4.91e3T^{2} \) |
| 19 | \( 1 + 155. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 - 2.43e4T^{2} \) |
| 31 | \( 1 + 155. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 110T + 5.06e4T^{2} \) |
| 41 | \( 1 - 6.89e4T^{2} \) |
| 43 | \( 1 - 218. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 1.03e5T^{2} \) |
| 53 | \( 1 - 1.48e5T^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 + 182T + 2.26e5T^{2} \) |
| 67 | \( 1 + 654. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.19e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.09e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 5.71e5T^{2} \) |
| 89 | \( 1 - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.33e3T + 9.12e5T^{2} \) |
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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
−11.52925075223636367977274557246, −10.72874747690761892007084898356, −9.959624652498773104774836292666, −9.090726508523707183617641411658, −7.66425583564609232077092167683, −6.75966867381010102619316302029, −4.99119024932835093425875404787, −4.29923251272244425073223636634, −2.88471476683056850439371472394, −0.38550567449165372010776371804,
1.84805953450394109073391430639, 2.94909743387969823483662727294, 5.15931570002782166839203250713, 6.01083526005726888950259177745, 7.21140623894425760295039773128, 8.288385667591260325808733910968, 9.103640294605888181883488790002, 10.35749313763207394316265647467, 11.97080426651678527172442316758, 12.11404556454237517898715750579