| L(s) = 1 | + (−1.60 + 1.85i)2-s + (5.95 − 3.82i)3-s + (0.281 + 1.96i)4-s + (2.81 + 6.16i)5-s + (−2.47 + 17.1i)6-s + (8.67 − 2.54i)7-s + (−20.5 − 13.2i)8-s + (9.61 − 21.0i)9-s + (−15.9 − 4.68i)10-s + (−42.0 − 48.4i)11-s + (9.18 + 10.6i)12-s + (−4.68 − 1.37i)13-s + (−9.21 + 20.1i)14-s + (40.3 + 25.9i)15-s + (42.4 − 12.4i)16-s + (−15.5 + 108. i)17-s + ⋯ |
| L(s) = 1 | + (−0.568 + 0.655i)2-s + (1.14 − 0.736i)3-s + (0.0352 + 0.245i)4-s + (0.251 + 0.551i)5-s + (−0.168 + 1.17i)6-s + (0.468 − 0.137i)7-s + (−0.910 − 0.585i)8-s + (0.356 − 0.780i)9-s + (−0.504 − 0.148i)10-s + (−1.15 − 1.32i)11-s + (0.220 + 0.255i)12-s + (−0.100 − 0.0293i)13-s + (−0.175 + 0.385i)14-s + (0.694 + 0.446i)15-s + (0.663 − 0.194i)16-s + (−0.221 + 1.54i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.11513 + 0.344051i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11513 + 0.344051i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (63.2 + 90.3i)T \) |
| good | 2 | \( 1 + (1.60 - 1.85i)T + (-1.13 - 7.91i)T^{2} \) |
| 3 | \( 1 + (-5.95 + 3.82i)T + (11.2 - 24.5i)T^{2} \) |
| 5 | \( 1 + (-2.81 - 6.16i)T + (-81.8 + 94.4i)T^{2} \) |
| 7 | \( 1 + (-8.67 + 2.54i)T + (288. - 185. i)T^{2} \) |
| 11 | \( 1 + (42.0 + 48.4i)T + (-189. + 1.31e3i)T^{2} \) |
| 13 | \( 1 + (4.68 + 1.37i)T + (1.84e3 + 1.18e3i)T^{2} \) |
| 17 | \( 1 + (15.5 - 108. i)T + (-4.71e3 - 1.38e3i)T^{2} \) |
| 19 | \( 1 + (5.73 + 39.9i)T + (-6.58e3 + 1.93e3i)T^{2} \) |
| 29 | \( 1 + (15.4 - 107. i)T + (-2.34e4 - 6.87e3i)T^{2} \) |
| 31 | \( 1 + (-282. - 181. i)T + (1.23e4 + 2.70e4i)T^{2} \) |
| 37 | \( 1 + (-45.6 + 99.8i)T + (-3.31e4 - 3.82e4i)T^{2} \) |
| 41 | \( 1 + (-10.7 - 23.5i)T + (-4.51e4 + 5.20e4i)T^{2} \) |
| 43 | \( 1 + (180. - 115. i)T + (3.30e4 - 7.23e4i)T^{2} \) |
| 47 | \( 1 - 28.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + (88.1 - 25.8i)T + (1.25e5 - 8.04e4i)T^{2} \) |
| 59 | \( 1 + (-591. - 173. i)T + (1.72e5 + 1.11e5i)T^{2} \) |
| 61 | \( 1 + (430. + 276. i)T + (9.42e4 + 2.06e5i)T^{2} \) |
| 67 | \( 1 + (438. - 505. i)T + (-4.28e4 - 2.97e5i)T^{2} \) |
| 71 | \( 1 + (-679. + 784. i)T + (-5.09e4 - 3.54e5i)T^{2} \) |
| 73 | \( 1 + (16.0 + 111. i)T + (-3.73e5 + 1.09e5i)T^{2} \) |
| 79 | \( 1 + (-428. - 125. i)T + (4.14e5 + 2.66e5i)T^{2} \) |
| 83 | \( 1 + (116. - 254. i)T + (-3.74e5 - 4.32e5i)T^{2} \) |
| 89 | \( 1 + (-388. + 249. i)T + (2.92e5 - 6.41e5i)T^{2} \) |
| 97 | \( 1 + (96.3 + 210. i)T + (-5.97e5 + 6.89e5i)T^{2} \) |
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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
−17.73632499506557539343343998308, −16.27568755372602311125500039854, −14.94770067040459847462021565937, −13.83664884155975243135245697504, −12.69875082320339694394046655428, −10.63286467471020369521600853811, −8.545608601002161262861345772133, −8.027340815605161606897063323764, −6.50216326849126192351697115303, −2.90228229356515639786211958242,
2.39305056268864227833153854555, 4.95583583852112767993071580772, 8.041519256036386198399918997803, 9.439667270371181570196042120971, 10.04434124697959658074338701964, 11.79134066937501893277205651613, 13.56815281937491704407379467504, 14.90584500974245044823894217242, 15.68131565751281142028957672174, 17.58403670381790775474637103987
