| L(s) = 1 | − 0.178·3-s − 3.28i·5-s + (1.92 + 1.81i)7-s − 2.96·9-s + 6.46i·11-s + i·13-s + 0.585i·15-s + 2.78i·17-s − 4.83·19-s + (−0.344 − 0.322i)21-s − 5.26i·23-s − 5.76·25-s + 1.06·27-s − 7.64·29-s − 3.85·31-s + ⋯ |
| L(s) = 1 | − 0.102·3-s − 1.46i·5-s + (0.729 + 0.684i)7-s − 0.989·9-s + 1.95i·11-s + 0.277i·13-s + 0.151i·15-s + 0.674i·17-s − 1.10·19-s + (−0.0750 − 0.0704i)21-s − 1.09i·23-s − 1.15·25-s + 0.204·27-s − 1.41·29-s − 0.693·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8106641384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8106641384\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-1.92 - 1.81i)T \) |
| 13 | \( 1 - iT \) |
| good | 3 | \( 1 + 0.178T + 3T^{2} \) |
| 5 | \( 1 + 3.28iT - 5T^{2} \) |
| 11 | \( 1 - 6.46iT - 11T^{2} \) |
| 17 | \( 1 - 2.78iT - 17T^{2} \) |
| 19 | \( 1 + 4.83T + 19T^{2} \) |
| 23 | \( 1 + 5.26iT - 23T^{2} \) |
| 29 | \( 1 + 7.64T + 29T^{2} \) |
| 31 | \( 1 + 3.85T + 31T^{2} \) |
| 37 | \( 1 + 5.94T + 37T^{2} \) |
| 41 | \( 1 - 12.4iT - 41T^{2} \) |
| 43 | \( 1 - 5.25iT - 43T^{2} \) |
| 47 | \( 1 - 8.67T + 47T^{2} \) |
| 53 | \( 1 - 4.64T + 53T^{2} \) |
| 59 | \( 1 + 3.68T + 59T^{2} \) |
| 61 | \( 1 + 4.84iT - 61T^{2} \) |
| 67 | \( 1 - 8.91iT - 67T^{2} \) |
| 71 | \( 1 - 9.88iT - 71T^{2} \) |
| 73 | \( 1 + 9.29iT - 73T^{2} \) |
| 79 | \( 1 + 1.03iT - 79T^{2} \) |
| 83 | \( 1 + 5.10T + 83T^{2} \) |
| 89 | \( 1 - 3.16iT - 89T^{2} \) |
| 97 | \( 1 + 13.0iT - 97T^{2} \) |
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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
−9.506799559327184142121816579000, −8.859818182356638389397242478764, −8.366798663538189931666130092796, −7.53270105673212862178122967376, −6.34753634962123826513904517675, −5.44640388231657879482999343125, −4.74554399321135925767489027412, −4.16042968113929958141901565734, −2.34153200290066275878310209510, −1.60163638723241685007652789672,
0.31042332315076434869544992331, 2.16818664415706791927453593386, 3.28675902580717033660672452071, 3.78714603007062012441524484937, 5.46502320919275785828138928679, 5.84423480140166712831459988975, 6.96098651555178829786399054975, 7.54475943789500380749936980276, 8.482523440313834775177692956011, 9.124999168639364036312972824833
