| L(s) = 1 | − 10.3i·5-s + 14.6·7-s − 5.65i·11-s + 17·25-s − 218. i·29-s + 338.·31-s − 152. i·35-s − 127·49-s + 509. i·53-s − 58.7·55-s − 554. i·59-s + 322·73-s − 83.1i·77-s − 308.·79-s − 1.22e3i·83-s + ⋯ |
| L(s) = 1 | − 0.929i·5-s + 0.793·7-s − 0.155i·11-s + 0.136·25-s − 1.39i·29-s + 1.95·31-s − 0.737i·35-s − 0.370·49-s + 1.31i·53-s − 0.144·55-s − 1.22i·59-s + 0.516·73-s − 0.123i·77-s − 0.439·79-s − 1.62i·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.159717860\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.159717860\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 10.3iT - 125T^{2} \) |
| 7 | \( 1 - 14.6T + 343T^{2} \) |
| 11 | \( 1 + 5.65iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 2.19e3T^{2} \) |
| 17 | \( 1 + 4.91e3T^{2} \) |
| 19 | \( 1 - 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 + 218. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 338.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 5.06e4T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 - 7.95e4T^{2} \) |
| 47 | \( 1 + 1.03e5T^{2} \) |
| 53 | \( 1 - 509. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 554. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 2.26e5T^{2} \) |
| 67 | \( 1 - 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 - 322T + 3.89e5T^{2} \) |
| 79 | \( 1 + 308.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.22e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 + 574T + 9.12e5T^{2} \) |
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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
−9.155455224758184372707373944141, −8.277289847687472496306383329577, −7.892468731485976170532611583115, −6.66342145460443092164477007017, −5.72692000074153191587926859085, −4.80054530558251278486492659496, −4.23314817664151869032207861889, −2.81056291967250817191022817103, −1.57775885256110712766981083427, −0.57284093945999394678194049924,
1.16599054570433810125257267172, 2.39697541741472857782276049880, 3.32033069383763134474198859438, 4.48461344444376810290787022169, 5.31796079151300375996184618699, 6.45792741759169404558737740631, 7.06503638499736935571269922608, 7.997752287957962528134853010532, 8.684377754583973868821292268741, 9.766676182498727691167429644801
