| L(s) = 1 | − i·2-s − 4-s − 2.23i·5-s + 2i·7-s + i·8-s − 2.23·10-s − 5.23·11-s + 4.47i·13-s + 2·14-s + 16-s − 4.47i·17-s + 7.23·19-s + 2.23i·20-s + 5.23i·22-s + 4i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.999i·5-s + 0.755i·7-s + 0.353i·8-s − 0.707·10-s − 1.57·11-s + 1.24i·13-s + 0.534·14-s + 0.250·16-s − 1.08i·17-s + 1.66·19-s + 0.499i·20-s + 1.11i·22-s + 0.834i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.420816441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.420816441\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 37 | \( 1 - iT \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 5.23T + 11T^{2} \) |
| 13 | \( 1 - 4.47iT - 13T^{2} \) |
| 17 | \( 1 + 4.47iT - 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + 2.47T + 31T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 + 10.4iT - 43T^{2} \) |
| 47 | \( 1 + 9.70iT - 47T^{2} \) |
| 53 | \( 1 - 12.4iT - 53T^{2} \) |
| 59 | \( 1 - 6.94T + 59T^{2} \) |
| 61 | \( 1 - 5.23T + 61T^{2} \) |
| 67 | \( 1 + 10.4iT - 67T^{2} \) |
| 71 | \( 1 - 2.47T + 71T^{2} \) |
| 73 | \( 1 + 10.4iT - 73T^{2} \) |
| 79 | \( 1 - 2.47T + 79T^{2} \) |
| 83 | \( 1 + 15.4iT - 83T^{2} \) |
| 89 | \( 1 + 7.52T + 89T^{2} \) |
| 97 | \( 1 + 8.18iT - 97T^{2} \) |
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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
−8.742583358941832234447544254885, −7.75705296179486079336464154616, −7.21726788708239249741021525553, −5.78973490497174609946241737075, −5.22357961710516906834780620057, −4.75483743473493346713781054592, −3.62297148162441168103154543617, −2.65306684101555539982342420554, −1.86166759101041526326162381819, −0.59691103476414835289244080956,
0.837008822747586868315898469466, 2.58306430623257398393348543256, 3.24857849580034438004674644354, 4.19264849169388590343702379301, 5.26813406530271218581767132734, 5.77052819544889488138933863213, 6.68292666642709742407372940676, 7.41144431142396458122593068636, 7.919602257844620690474166662780, 8.395329708366328418594556904941
