| L(s) = 1 | − 9.32i·3-s − 14.1·5-s + (−3.84 − 18.1i)7-s − 59.9·9-s − 16.0i·11-s + 65.9i·13-s + 131. i·15-s − 10.3·17-s − 61.4·19-s + (−168. + 35.9i)21-s + (−22.0 − 108. i)23-s + 74.3·25-s + 307. i·27-s + 108.·29-s + 306. i·31-s + ⋯ |
| L(s) = 1 | − 1.79i·3-s − 1.26·5-s + (−0.207 − 0.978i)7-s − 2.22·9-s − 0.440i·11-s + 1.40i·13-s + 2.26i·15-s − 0.148·17-s − 0.742·19-s + (−1.75 + 0.373i)21-s + (−0.200 − 0.979i)23-s + 0.594·25-s + 2.19i·27-s + 0.692·29-s + 1.77i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.399i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.916 - 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1848147746\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1848147746\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (3.84 + 18.1i)T \) |
| 23 | \( 1 + (22.0 + 108. i)T \) |
| good | 3 | \( 1 + 9.32iT - 27T^{2} \) |
| 5 | \( 1 + 14.1T + 125T^{2} \) |
| 11 | \( 1 + 16.0iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 65.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 10.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 61.4T + 6.85e3T^{2} \) |
| 29 | \( 1 - 108.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 306. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 320. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 341. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 63.4iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 96.6iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 83.2iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 470. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 63.3T + 2.26e5T^{2} \) |
| 67 | \( 1 - 286. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 150.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 569. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 334. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 780.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 636.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.19e3T + 9.12e5T^{2} \) |
| 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
−10.59692463077683076184559559866, −8.860312301977714474912453299745, −8.346937494226615543242150836309, −7.30725630386579330044707532968, −6.98104771922943150386552483945, −6.15865416765754068499691499746, −4.51390286042891834786435670659, −3.52248020610736727661605456922, −2.13276203637284128584586187435, −0.853590488568972253495323876283,
0.07069189423891561443734348160, 2.75083746107861330434474677727, 3.57178190884550948391071280399, 4.46820226459414320981624722304, 5.26407104542391764692407254238, 6.24481444462424173680620705674, 7.952334124177887651080753782644, 8.325359663902783146745246925684, 9.472188064686777275329490611742, 9.970233014947111295499030816492
