| L(s) = 1 | − 0.719i·2-s − 2.00i·3-s + 1.48·4-s − 1.44·6-s + 3.86i·7-s − 2.50i·8-s − 1.03·9-s + 1.66·11-s − 2.97i·12-s + 3.40i·13-s + 2.78·14-s + 1.16·16-s + 5.85i·17-s + 0.746i·18-s + 7.21·19-s + ⋯ |
| L(s) = 1 | − 0.508i·2-s − 1.16i·3-s + 0.741·4-s − 0.590·6-s + 1.46i·7-s − 0.885i·8-s − 0.346·9-s + 0.502·11-s − 0.859i·12-s + 0.943i·13-s + 0.743·14-s + 0.290·16-s + 1.41i·17-s + 0.176i·18-s + 1.65·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.219064271\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.219064271\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 47 | \( 1 + iT \) |
| good | 2 | \( 1 + 0.719iT - 2T^{2} \) |
| 3 | \( 1 + 2.00iT - 3T^{2} \) |
| 7 | \( 1 - 3.86iT - 7T^{2} \) |
| 11 | \( 1 - 1.66T + 11T^{2} \) |
| 13 | \( 1 - 3.40iT - 13T^{2} \) |
| 17 | \( 1 - 5.85iT - 17T^{2} \) |
| 19 | \( 1 - 7.21T + 19T^{2} \) |
| 23 | \( 1 + 7.77iT - 23T^{2} \) |
| 29 | \( 1 + 6.47T + 29T^{2} \) |
| 31 | \( 1 - 8.70T + 31T^{2} \) |
| 37 | \( 1 + 4.86iT - 37T^{2} \) |
| 41 | \( 1 + 3.00T + 41T^{2} \) |
| 43 | \( 1 - 3.70iT - 43T^{2} \) |
| 53 | \( 1 - 11.9iT - 53T^{2} \) |
| 59 | \( 1 + 9.37T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 - 11.2iT - 67T^{2} \) |
| 71 | \( 1 + 6.59T + 71T^{2} \) |
| 73 | \( 1 + 13.9iT - 73T^{2} \) |
| 79 | \( 1 + 3.75T + 79T^{2} \) |
| 83 | \( 1 + 2.26iT - 83T^{2} \) |
| 89 | \( 1 - 1.47T + 89T^{2} \) |
| 97 | \( 1 + 1.96iT - 97T^{2} \) |
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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
−9.606799738406780852720495299206, −8.795118843101252193391470075146, −7.929958105778692934076010568640, −7.04748754677149186358928853343, −6.31162602699128022796692316556, −5.79614695776206150841060069059, −4.25746911646399320963845976426, −2.93587819547454542424870791570, −2.09222194509178993828716506625, −1.31906500400172444369987913226,
1.18339389643562249378078584598, 3.09906251500100854620823744866, 3.70896856385010153007444380529, 4.94642662587724929029769862202, 5.47557312517098463452136606723, 6.80972853722979141441199498243, 7.38150750894708530012779554128, 8.015294667133239876505246268456, 9.425889246285129843544765349217, 9.882194193472291491111661660885
