| L(s) = 1 | + (−1.87 + 0.684i)2-s + (6.95 + 2.53i)3-s + (3.06 − 2.57i)4-s + (2.65 + 15.0i)5-s − 14.8·6-s + (−1.00 − 5.69i)7-s + (−4.00 + 6.92i)8-s + (21.2 + 17.8i)9-s + (−15.2 − 26.4i)10-s + (19.0 − 33.0i)11-s + (27.8 − 10.1i)12-s + (−55.8 + 46.8i)13-s + (5.78 + 10.0i)14-s + (−19.6 + 111. i)15-s + (2.77 − 15.7i)16-s + (91.0 + 76.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 + 0.241i)2-s + (1.33 + 0.487i)3-s + (0.383 − 0.321i)4-s + (0.237 + 1.34i)5-s − 1.00·6-s + (−0.0542 − 0.307i)7-s + (−0.176 + 0.306i)8-s + (0.787 + 0.661i)9-s + (−0.483 − 0.837i)10-s + (0.522 − 0.905i)11-s + (0.669 − 0.243i)12-s + (−1.19 + 1.00i)13-s + (0.110 + 0.191i)14-s + (−0.338 + 1.91i)15-s + (0.0434 − 0.246i)16-s + (1.29 + 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.31263 + 1.01468i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31263 + 1.01468i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.87 - 0.684i)T \) |
| 37 | \( 1 + (108. + 197. i)T \) |
| good | 3 | \( 1 + (-6.95 - 2.53i)T + (20.6 + 17.3i)T^{2} \) |
| 5 | \( 1 + (-2.65 - 15.0i)T + (-117. + 42.7i)T^{2} \) |
| 7 | \( 1 + (1.00 + 5.69i)T + (-322. + 117. i)T^{2} \) |
| 11 | \( 1 + (-19.0 + 33.0i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (55.8 - 46.8i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-91.0 - 76.3i)T + (853. + 4.83e3i)T^{2} \) |
| 19 | \( 1 + (-24.7 - 8.99i)T + (5.25e3 + 4.40e3i)T^{2} \) |
| 23 | \( 1 + (62.8 + 108. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-93.1 + 161. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 - 76.2T + 2.97e4T^{2} \) |
| 41 | \( 1 + (-32.1 + 27.0i)T + (1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 - 83.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + (286. + 496. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (50.3 - 285. i)T + (-1.39e5 - 5.09e4i)T^{2} \) |
| 59 | \( 1 + (-49.2 + 279. i)T + (-1.92e5 - 7.02e4i)T^{2} \) |
| 61 | \( 1 + (312. - 262. i)T + (3.94e4 - 2.23e5i)T^{2} \) |
| 67 | \( 1 + (-48.3 - 274. i)T + (-2.82e5 + 1.02e5i)T^{2} \) |
| 71 | \( 1 + (-934. - 340. i)T + (2.74e5 + 2.30e5i)T^{2} \) |
| 73 | \( 1 + 970.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (80.9 + 458. i)T + (-4.63e5 + 1.68e5i)T^{2} \) |
| 83 | \( 1 + (-402. - 337. i)T + (9.92e4 + 5.63e5i)T^{2} \) |
| 89 | \( 1 + (-118. + 672. i)T + (-6.62e5 - 2.41e5i)T^{2} \) |
| 97 | \( 1 + (259. + 450. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−14.36472260026471144683467179359, −13.97095230320401874922823625705, −11.88616851202493959496504399538, −10.41680018520452420950944526425, −9.828691209474030704880221116413, −8.604979756000057030423722993487, −7.50843281139120201390094769844, −6.29015760026040286623878541512, −3.75135868594774455434959745310, −2.42641527752639436639712312575,
1.36440966816401108926293800723, 2.93979872780500281195450580725, 5.09344134969635783338889274716, 7.37260268492111290219276014972, 8.155530856455167030337352049429, 9.333492978906849906164771721676, 9.794681775584445677198750655965, 12.11127107035414107504609288003, 12.56332812733328673939183090076, 13.78662059559783657743057280353
