| L(s) = 1 | + (−1.87 − 0.684i)2-s + (3.06 + 2.57i)4-s + (2.36 − 13.4i)5-s + (−10.7 + 8.98i)7-s + (−4.00 − 6.92i)8-s + (−13.6 + 23.6i)10-s + (−5.21 − 29.5i)11-s + (9.46 − 3.44i)13-s + (26.2 − 9.56i)14-s + (2.77 + 15.7i)16-s + (−45.9 + 79.6i)17-s + (−21.7 − 37.7i)19-s + (41.7 − 35.0i)20-s + (−10.4 + 59.1i)22-s + (−85.9 − 72.1i)23-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (0.383 + 0.321i)4-s + (0.211 − 1.20i)5-s + (−0.578 + 0.485i)7-s + (−0.176 − 0.306i)8-s + (−0.431 + 0.746i)10-s + (−0.143 − 0.811i)11-s + (0.201 − 0.0735i)13-s + (0.501 − 0.182i)14-s + (0.0434 + 0.246i)16-s + (−0.655 + 1.13i)17-s + (−0.263 − 0.455i)19-s + (0.467 − 0.391i)20-s + (−0.101 + 0.573i)22-s + (−0.779 − 0.653i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0733i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0163092 - 0.444321i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0163092 - 0.444321i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.87 + 0.684i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-2.36 + 13.4i)T + (-117. - 42.7i)T^{2} \) |
| 7 | \( 1 + (10.7 - 8.98i)T + (59.5 - 337. i)T^{2} \) |
| 11 | \( 1 + (5.21 + 29.5i)T + (-1.25e3 + 455. i)T^{2} \) |
| 13 | \( 1 + (-9.46 + 3.44i)T + (1.68e3 - 1.41e3i)T^{2} \) |
| 17 | \( 1 + (45.9 - 79.6i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (21.7 + 37.7i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (85.9 + 72.1i)T + (2.11e3 + 1.19e4i)T^{2} \) |
| 29 | \( 1 + (271. + 98.9i)T + (1.86e4 + 1.56e4i)T^{2} \) |
| 31 | \( 1 + (142. + 119. i)T + (5.17e3 + 2.93e4i)T^{2} \) |
| 37 | \( 1 + (195. - 338. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (345. - 125. i)T + (5.27e4 - 4.43e4i)T^{2} \) |
| 43 | \( 1 + (32.5 + 184. i)T + (-7.47e4 + 2.71e4i)T^{2} \) |
| 47 | \( 1 + (-257. + 216. i)T + (1.80e4 - 1.02e5i)T^{2} \) |
| 53 | \( 1 - 366.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-65.6 + 372. i)T + (-1.92e5 - 7.02e4i)T^{2} \) |
| 61 | \( 1 + (-675. + 567. i)T + (3.94e4 - 2.23e5i)T^{2} \) |
| 67 | \( 1 + (-897. + 326. i)T + (2.30e5 - 1.93e5i)T^{2} \) |
| 71 | \( 1 + (341. - 590. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (410. + 710. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (132. + 48.1i)T + (3.77e5 + 3.16e5i)T^{2} \) |
| 83 | \( 1 + (-241. - 87.7i)T + (4.38e5 + 3.67e5i)T^{2} \) |
| 89 | \( 1 + (-470. - 814. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-40.0 - 226. i)T + (-8.57e5 + 3.12e5i)T^{2} \) |
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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
−11.91046502604428511843028738167, −10.84285570658045270883472746406, −9.706403273303694802951973303154, −8.771302528760882717417888184764, −8.228358230472238469933337132591, −6.51641901397883261705141812397, −5.43831260760108785719209264706, −3.78078146868037946691393471114, −1.97972092515402098139848216501, −0.24125716690543399684394298522,
2.13546464777809566455611572636, 3.66376467477573464336594709332, 5.60475840417191770198365327367, 7.02682897975937108255380247723, 7.23973029907399760526974810070, 8.952369920978618228627287854972, 9.949832881554181584798868883067, 10.63075715202270461511857616367, 11.60153919588872601337274987357, 12.94282161398077499217533507215
