| L(s) = 1 | − 3-s + 5-s + 0.238i·7-s + 9-s + 5.02i·11-s + 5.35i·13-s − 15-s + 4.36·17-s + (1.72 − 4.00i)19-s − 0.238i·21-s + 5.83i·23-s + 25-s − 27-s + 8.48i·29-s − 5.65·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.0900i·7-s + 0.333·9-s + 1.51i·11-s + 1.48i·13-s − 0.258·15-s + 1.05·17-s + (0.395 − 0.918i)19-s − 0.0519i·21-s + 1.21i·23-s + 0.200·25-s − 0.192·27-s + 1.57i·29-s − 1.01·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.330965643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.330965643\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + (-1.72 + 4.00i)T \) |
| good | 7 | \( 1 - 0.238iT - 7T^{2} \) |
| 11 | \( 1 - 5.02iT - 11T^{2} \) |
| 13 | \( 1 - 5.35iT - 13T^{2} \) |
| 17 | \( 1 - 4.36T + 17T^{2} \) |
| 23 | \( 1 - 5.83iT - 23T^{2} \) |
| 29 | \( 1 - 8.48iT - 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 - 6.51iT - 37T^{2} \) |
| 41 | \( 1 + 5.59iT - 41T^{2} \) |
| 43 | \( 1 + 6.99iT - 43T^{2} \) |
| 47 | \( 1 + 0.328iT - 47T^{2} \) |
| 53 | \( 1 + 9.29iT - 53T^{2} \) |
| 59 | \( 1 + 1.71T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 + 3.29T + 71T^{2} \) |
| 73 | \( 1 - 8.74T + 73T^{2} \) |
| 79 | \( 1 + 7.52T + 79T^{2} \) |
| 83 | \( 1 - 7.08iT - 83T^{2} \) |
| 89 | \( 1 - 5.67iT - 89T^{2} \) |
| 97 | \( 1 + 10.4iT - 97T^{2} \) |
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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
−8.850391161629593462632839866073, −7.46624755550102590957397609545, −7.17658708056922693019516873465, −6.52272130833129601082105792419, −5.46862408427757536059398744901, −5.04965619487656702218474377925, −4.21721292656577784926038306805, −3.26715114746539053474085660200, −1.99659966350430822472974890173, −1.41105921582217648055880046634,
0.42198887051908162004788314493, 1.28357170064498520238122638004, 2.72339314773738412627209365275, 3.38306091838586693668106893980, 4.35851969298134272319383308816, 5.42777173025561402010324776817, 5.91226114081421053281779257070, 6.21064583836727484908222900600, 7.63060459093351042697398992560, 7.84579793873147185434279927755
