| L(s) = 1 | + i·3-s − i·5-s − 2·7-s − 9-s + 1.46i·11-s − 1.46i·13-s + 15-s − 3.46·17-s + 6.92i·19-s − 2i·21-s − 4·23-s − 25-s − i·27-s + 4.92i·29-s − 7.46·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.447i·5-s − 0.755·7-s − 0.333·9-s + 0.441i·11-s − 0.406i·13-s + 0.258·15-s − 0.840·17-s + 1.58i·19-s − 0.436i·21-s − 0.834·23-s − 0.200·25-s − 0.192i·27-s + 0.915i·29-s − 1.34·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0561921 + 0.426821i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0561921 + 0.426821i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + iT \) |
| good | 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 1.46iT - 11T^{2} \) |
| 13 | \( 1 + 1.46iT - 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 6.92iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 4.92iT - 29T^{2} \) |
| 31 | \( 1 + 7.46T + 31T^{2} \) |
| 37 | \( 1 + 2.53iT - 37T^{2} \) |
| 41 | \( 1 + 8.92T + 41T^{2} \) |
| 43 | \( 1 - 6.92iT - 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 + 12.9iT - 53T^{2} \) |
| 59 | \( 1 - 2.53iT - 59T^{2} \) |
| 61 | \( 1 - 4iT - 61T^{2} \) |
| 67 | \( 1 - 6.92iT - 67T^{2} \) |
| 71 | \( 1 + 6.92T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 6.39T + 79T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 0.928T + 97T^{2} \) |
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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
−10.19485394336769475921239115015, −9.708826610739511848773486215761, −8.823961227443358454416612792019, −8.057002644153694982699164863221, −6.99121605412110690812953763470, −6.00275596419484365203390871301, −5.20202905090805955754151396484, −4.11004329381752899349374828778, −3.33007010539779106782583254286, −1.86095555405810258373922402236,
0.18572842624847487243850789688, 2.06663045020158302442231631586, 3.06127327217362228335623237215, 4.16650753041499254463902367224, 5.45019366981643563032793904104, 6.49068351820525679154532060251, 6.88227941416571216392276642571, 7.88792197553860943065143877808, 8.898957335225494009222580641807, 9.503994028980192511300549039624
