| L(s) = 1 | + (−1.30 + 1.51i)2-s + (−2.52 + 1.62i)3-s + (−0.569 − 3.95i)4-s + (−3.80 − 8.32i)5-s + (0.853 − 5.93i)6-s + (2.38 − 0.699i)7-s + (6.73 + 4.32i)8-s + (3.73 − 8.18i)9-s + (17.5 + 5.15i)10-s + (36.6 + 42.3i)11-s + (7.85 + 9.06i)12-s + (−6.64 − 1.94i)13-s + (−2.06 + 4.51i)14-s + (23.0 + 14.8i)15-s + (−15.3 + 4.50i)16-s + (−5.76 + 40.1i)17-s + ⋯ |
| L(s) = 1 | + (−0.463 + 0.534i)2-s + (−0.485 + 0.312i)3-s + (−0.0711 − 0.494i)4-s + (−0.339 − 0.744i)5-s + (0.0580 − 0.404i)6-s + (0.128 − 0.0377i)7-s + (0.297 + 0.191i)8-s + (0.138 − 0.303i)9-s + (0.555 + 0.162i)10-s + (1.00 + 1.16i)11-s + (0.189 + 0.218i)12-s + (−0.141 − 0.0415i)13-s + (−0.0393 + 0.0862i)14-s + (0.397 + 0.255i)15-s + (−0.239 + 0.0704i)16-s + (−0.0822 + 0.572i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0157 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0157 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.673802 + 0.663251i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.673802 + 0.663251i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.30 - 1.51i)T \) |
| 3 | \( 1 + (2.52 - 1.62i)T \) |
| 23 | \( 1 + (16.1 - 109. i)T \) |
| good | 5 | \( 1 + (3.80 + 8.32i)T + (-81.8 + 94.4i)T^{2} \) |
| 7 | \( 1 + (-2.38 + 0.699i)T + (288. - 185. i)T^{2} \) |
| 11 | \( 1 + (-36.6 - 42.3i)T + (-189. + 1.31e3i)T^{2} \) |
| 13 | \( 1 + (6.64 + 1.94i)T + (1.84e3 + 1.18e3i)T^{2} \) |
| 17 | \( 1 + (5.76 - 40.1i)T + (-4.71e3 - 1.38e3i)T^{2} \) |
| 19 | \( 1 + (-3.74 - 26.0i)T + (-6.58e3 + 1.93e3i)T^{2} \) |
| 29 | \( 1 + (38.6 - 268. i)T + (-2.34e4 - 6.87e3i)T^{2} \) |
| 31 | \( 1 + (-136. - 87.7i)T + (1.23e4 + 2.70e4i)T^{2} \) |
| 37 | \( 1 + (-62.9 + 137. i)T + (-3.31e4 - 3.82e4i)T^{2} \) |
| 41 | \( 1 + (-149. - 326. i)T + (-4.51e4 + 5.20e4i)T^{2} \) |
| 43 | \( 1 + (-263. + 169. i)T + (3.30e4 - 7.23e4i)T^{2} \) |
| 47 | \( 1 - 132.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-550. + 161. i)T + (1.25e5 - 8.04e4i)T^{2} \) |
| 59 | \( 1 + (230. + 67.7i)T + (1.72e5 + 1.11e5i)T^{2} \) |
| 61 | \( 1 + (308. + 197. i)T + (9.42e4 + 2.06e5i)T^{2} \) |
| 67 | \( 1 + (82.0 - 94.7i)T + (-4.28e4 - 2.97e5i)T^{2} \) |
| 71 | \( 1 + (-539. + 623. i)T + (-5.09e4 - 3.54e5i)T^{2} \) |
| 73 | \( 1 + (-30.7 - 213. i)T + (-3.73e5 + 1.09e5i)T^{2} \) |
| 79 | \( 1 + (-49.8 - 14.6i)T + (4.14e5 + 2.66e5i)T^{2} \) |
| 83 | \( 1 + (55.8 - 122. i)T + (-3.74e5 - 4.32e5i)T^{2} \) |
| 89 | \( 1 + (646. - 415. i)T + (2.92e5 - 6.41e5i)T^{2} \) |
| 97 | \( 1 + (200. + 440. i)T + (-5.97e5 + 6.89e5i)T^{2} \) |
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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
−12.70964878454821800998719887091, −12.01341240327685974135901716318, −10.80784540834703069472435580984, −9.681087207618329271272514256204, −8.867515976183881171689404067469, −7.62518561912204614249826110953, −6.50728008300261727113831866450, −5.15471418738797223988777643779, −4.11764077617532205797265472892, −1.32512547346533762984223858857,
0.68163938915433247067743071202, 2.67607487837025488319460342843, 4.17495971532938153441751639495, 6.03871609760275656634685428786, 7.09097683178378758417217902396, 8.261213743523575164972001694314, 9.428351487363133710168336724883, 10.68646087427203354968717964988, 11.42727082191136224878737479776, 12.04193354573288571435097654054
