| L(s) = 1 | − 1.41i·2-s + (−2.37 + 1.83i)3-s − 2.00·4-s − 0.546i·5-s + (2.59 + 3.35i)6-s + 0.308·7-s + 2.82i·8-s + (2.28 − 8.70i)9-s − 0.772·10-s − 7.87i·11-s + (4.74 − 3.66i)12-s − 8.39·13-s − 0.436i·14-s + (1.00 + 1.29i)15-s + 4.00·16-s + 32.6i·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + (−0.791 + 0.610i)3-s − 0.500·4-s − 0.109i·5-s + (0.432 + 0.559i)6-s + 0.0440·7-s + 0.353i·8-s + (0.253 − 0.967i)9-s − 0.0772·10-s − 0.715i·11-s + (0.395 − 0.305i)12-s − 0.645·13-s − 0.0311i·14-s + (0.0667 + 0.0864i)15-s + 0.250·16-s + 1.92i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.610i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.791 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.922984 + 0.314743i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.922984 + 0.314743i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 + (2.37 - 1.83i)T \) |
| 59 | \( 1 - 7.68iT \) |
| good | 5 | \( 1 + 0.546iT - 25T^{2} \) |
| 7 | \( 1 - 0.308T + 49T^{2} \) |
| 11 | \( 1 + 7.87iT - 121T^{2} \) |
| 13 | \( 1 + 8.39T + 169T^{2} \) |
| 17 | \( 1 - 32.6iT - 289T^{2} \) |
| 19 | \( 1 - 13.0T + 361T^{2} \) |
| 23 | \( 1 - 29.5iT - 529T^{2} \) |
| 29 | \( 1 - 19.6iT - 841T^{2} \) |
| 31 | \( 1 - 7.71T + 961T^{2} \) |
| 37 | \( 1 - 5.62T + 1.36e3T^{2} \) |
| 41 | \( 1 + 23.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 80.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 70.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 69.7iT - 2.80e3T^{2} \) |
| 61 | \( 1 + 96.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 94.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + 10.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 10.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 92.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 101. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 135. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 108.T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.06507047531931009376221845964, −10.73465440887102138819103257232, −9.663130212403804223690880554842, −8.913173898021366456050204167592, −7.67257427615597012900231681187, −6.20375938149500934883034186427, −5.34719295015468480628132436393, −4.23575475431146157121403557622, −3.18583598696257350425829205383, −1.22412551427436503338827219331,
0.57762718006624377914768359828, 2.54670395724000293689137346656, 4.62868431466430274342587791114, 5.24100730065458836425168044774, 6.55766532676843557214803550339, 7.16812545434804966499216788428, 7.977071543609925878167271071056, 9.324853312025640845218302426633, 10.16051217708498943519295626457, 11.31264588404376036983729403908
