| L(s) = 1 | + 17.4i·5-s + 2.99·7-s + 10.6i·11-s − 43.3i·13-s + 37.8·17-s − 79.8i·19-s − 191.·23-s − 178.·25-s − 138. i·29-s − 212.·31-s + 52.1i·35-s + 270. i·37-s + 441.·41-s + 64.1i·43-s − 436.·47-s + ⋯ |
| L(s) = 1 | + 1.55i·5-s + 0.161·7-s + 0.291i·11-s − 0.924i·13-s + 0.540·17-s − 0.964i·19-s − 1.73·23-s − 1.43·25-s − 0.889i·29-s − 1.22·31-s + 0.251i·35-s + 1.20i·37-s + 1.68·41-s + 0.227i·43-s − 1.35·47-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
| \(L(2)\) |
\(\approx\) |
\(0.7693680781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7693680781\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 17.4iT - 125T^{2} \) |
| 7 | \( 1 - 2.99T + 343T^{2} \) |
| 11 | \( 1 - 10.6iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 43.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 37.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 79.8iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 191.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 138. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 212.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 270. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 441.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 64.1iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 436.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 278. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 830. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 724. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 859. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 681.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 785.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.01e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 467. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 510.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 234.T + 9.12e5T^{2} \) |
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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
−9.545390623722643941518302681208, −8.016483398229960459113594631876, −7.71442614369701921979922180986, −6.62922633347270562661977082128, −6.06038209565909892675570533451, −4.94544536973608910052291060768, −3.71310039081462585177292609242, −2.92298887870634147017861010240, −1.95623543902953692579732440371, −0.18704147047753559682033196515,
1.16329946430623431297983349548, 2.01149314010461355207566147560, 3.73383295787935081753443259970, 4.38382548397458244091819551848, 5.44082219097473619651159886351, 5.98955793832954602191244295529, 7.33380364390231259583236040319, 8.118771112546601554895089009731, 8.824665832507377112370772222208, 9.460108207073213029459648292145
