| L(s) = 1 | + 2.03i·3-s − 0.710i·5-s + 76.8·9-s − 151.·11-s − 260. i·13-s + 1.44·15-s + 385. i·17-s + 390. i·19-s + 177.·23-s + 624.·25-s + 321. i·27-s − 320.·29-s − 1.34e3i·31-s − 308. i·33-s − 797.·37-s + ⋯ |
| L(s) = 1 | + 0.225i·3-s − 0.0284i·5-s + 0.948·9-s − 1.25·11-s − 1.54i·13-s + 0.00642·15-s + 1.33i·17-s + 1.08i·19-s + 0.335·23-s + 0.999·25-s + 0.440i·27-s − 0.381·29-s − 1.40i·31-s − 0.283i·33-s − 0.582·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.966860260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.966860260\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2.03iT - 81T^{2} \) |
| 5 | \( 1 + 0.710iT - 625T^{2} \) |
| 11 | \( 1 + 151.T + 1.46e4T^{2} \) |
| 13 | \( 1 + 260. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 385. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 390. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 177.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 320.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.34e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 797.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 815. iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 2.16e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 4.28e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 3.17e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 4.70e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 2.53e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 4.09e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 2.25e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 4.65e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 4.19e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 7.79e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 9.46e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 9.94e3iT - 8.85e7T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.01678112773171057349010062281, −8.689039292793556652774962124456, −7.923145475612233738258697221851, −7.26853272404277271156211552111, −5.93554209011586964285843024726, −5.29437482348242038299326068045, −4.15220972280855422168972767826, −3.19812384611801695758997104056, −1.94086967683122657317748911191, −0.59098548470444958289279929571,
0.843133903940141995863356957210, 2.11149656247420737604142336600, 3.12395818048795462532171945189, 4.61866687242585342899273819597, 5.02553059523935995237408960216, 6.59714511164321484599541300574, 7.07290569442895689486268750151, 7.924378169016587750925780805837, 9.138182789158756071895298526414, 9.573751192151417222817811146073
