| L(s) = 1 | − 2-s + 4-s − 5-s + 2.64i·7-s − 8-s + 10-s − 1.34·11-s + 2.35i·13-s − 2.64i·14-s + 16-s + 6.24i·17-s + 3.10·19-s − 20-s + 1.34·22-s − 7.74i·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.997i·7-s − 0.353·8-s + 0.316·10-s − 0.405·11-s + 0.653i·13-s − 0.705i·14-s + 0.250·16-s + 1.51i·17-s + 0.712·19-s − 0.223·20-s + 0.286·22-s − 1.61i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.785 - 0.618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.785 - 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8429178918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8429178918\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + (0.424 - 8.17i)T \) |
| good | 7 | \( 1 - 2.64iT - 7T^{2} \) |
| 11 | \( 1 + 1.34T + 11T^{2} \) |
| 13 | \( 1 - 2.35iT - 13T^{2} \) |
| 17 | \( 1 - 6.24iT - 17T^{2} \) |
| 19 | \( 1 - 3.10T + 19T^{2} \) |
| 23 | \( 1 + 7.74iT - 23T^{2} \) |
| 29 | \( 1 - 8.46iT - 29T^{2} \) |
| 31 | \( 1 + 7.92iT - 31T^{2} \) |
| 37 | \( 1 - 9.02T + 37T^{2} \) |
| 41 | \( 1 + 7.85T + 41T^{2} \) |
| 43 | \( 1 - 10.3iT - 43T^{2} \) |
| 47 | \( 1 - 3.95iT - 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 - 5.88iT - 59T^{2} \) |
| 61 | \( 1 - 3.93iT - 61T^{2} \) |
| 71 | \( 1 + 8.97iT - 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + 2.89iT - 79T^{2} \) |
| 83 | \( 1 + 11.7iT - 83T^{2} \) |
| 89 | \( 1 - 9.61iT - 89T^{2} \) |
| 97 | \( 1 + 1.56iT - 97T^{2} \) |
| 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
−8.353593485745613606251086098835, −7.88041456705690728388152443427, −7.01077452755050322557333415969, −6.27156381696294637271947158991, −5.71359687301067281856672238114, −4.73615956862863055914049563215, −3.93304811815905566606645314533, −2.88071895701471891821313266969, −2.21631682657810565764648180629, −1.13680193858666955546389931784,
0.33530888302751174271265848960, 1.09011435733079169897522714607, 2.40227633037069467077136584669, 3.29198772632810550103692149401, 3.94397891180525203588200945341, 5.08129992927994189188574341982, 5.53976993682082704074388830071, 6.78424806211131381718979100703, 7.21565923672557307348150054223, 7.79329532998220209596535116371
