| L(s) = 1 | − i·2-s + i·3-s − 4-s + 2.23·5-s + 6-s − 4i·7-s + i·8-s − 9-s − 2.23i·10-s − i·12-s − 2.47i·13-s − 4·14-s + 2.23i·15-s + 16-s − 2.47i·17-s + i·18-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.999·5-s + 0.408·6-s − 1.51i·7-s + 0.353i·8-s − 0.333·9-s − 0.707i·10-s − 0.288i·12-s − 0.685i·13-s − 1.06·14-s + 0.577i·15-s + 0.250·16-s − 0.599i·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.11133 - 1.11133i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11133 - 1.11133i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 - 2.23T \) |
| 23 | \( 1 + iT \) |
| good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2.47iT - 13T^{2} \) |
| 17 | \( 1 + 2.47iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 29 | \( 1 - 0.472T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 0.472iT - 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 4.94iT - 47T^{2} \) |
| 53 | \( 1 + 8.94iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 0.472T + 61T^{2} \) |
| 67 | \( 1 - 4.94iT - 67T^{2} \) |
| 71 | \( 1 + 7.52T + 71T^{2} \) |
| 73 | \( 1 - 4.94iT - 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 - 1.52iT - 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 - 13.4iT - 97T^{2} \) |
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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
−10.35467123035456369017867859923, −9.703086645708493788404450337458, −8.838604748751368389923133691715, −7.72831721545751071895742683872, −6.66566162665793563392587713721, −5.51196372421299521537249703708, −4.58035063864447824964024719523, −3.63777736224836835194064270189, −2.46049794526921777214325559907, −0.875261983224079283209916285389,
1.73598765195195748507591192369, 2.77488469192435916606173335376, 4.53755850466110474288282416697, 5.77203142946814098564198949489, 6.02348120811327928931288557028, 7.01130481048423436630545642232, 8.148102206270035950746413973406, 8.973616268598580389643748324345, 9.411903395146954339411520683415, 10.59174772233478092958376820289
