| L(s) = 1 | − 3.53i·2-s − 13.5i·3-s + 19.5·4-s − 48·6-s − 49i·7-s − 181. i·8-s + 58.2·9-s − 691.·11-s − 265. i·12-s + 502. i·13-s − 173.·14-s − 17.5·16-s − 991. i·17-s − 205. i·18-s − 661.·19-s + ⋯ |
| L(s) = 1 | − 0.624i·2-s − 0.872i·3-s + 0.610·4-s − 0.544·6-s − 0.377i·7-s − 1.00i·8-s + 0.239·9-s − 1.72·11-s − 0.532i·12-s + 0.824i·13-s − 0.235·14-s − 0.0171·16-s − 0.831i·17-s − 0.149i·18-s − 0.420·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.411816471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.411816471\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + 49iT \) |
| good | 2 | \( 1 + 3.53iT - 32T^{2} \) |
| 3 | \( 1 + 13.5iT - 243T^{2} \) |
| 11 | \( 1 + 691.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 502. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 991. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 661.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.41e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 6.75e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.92e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 627. iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.62e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.72e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 4.29e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.59e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 8.90e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.89e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.25e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 1.89e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.01e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 9.69e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.07e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 4.24e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.04e5iT - 8.58e9T^{2} \) |
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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
−11.18895020726760335964572076042, −10.55707613432437313840759073504, −9.433102314605701852596091012904, −7.78324681559379128902718192488, −7.20948806183889104988766593049, −6.11532737770200459150185844173, −4.45210214488369582362756230663, −2.77483771216783465775175960640, −1.83226387416899574484958251428, −0.40208944417240909277229837993,
2.11283251759477412050170426700, 3.51894398908520082493430558655, 5.18842814626403583143181418779, 5.78100939040502500919961371548, 7.37695836599371731146134605929, 8.069338225052385317957791620713, 9.406424916860404758634111049778, 10.59398232408195578667002505720, 10.97073858117404018110842759673, 12.48927070421453238382968998664
