L(s) = 1 | + (−299. − 415. i)2-s + 5.93e3·3-s + (−8.25e4 + 2.48e5i)4-s + 2.28e6i·5-s + (−1.77e6 − 2.46e6i)6-s − 3.15e7i·7-s + (1.28e8 − 4.02e7i)8-s − 3.52e8·9-s + (9.47e8 − 6.83e8i)10-s + 2.35e9·11-s + (−4.90e8 + 1.47e9i)12-s − 1.20e10i·13-s + (−1.31e10 + 9.46e9i)14-s + 1.35e10i·15-s + (−5.50e10 − 4.10e10i)16-s − 1.97e11·17-s + ⋯ |
L(s) = 1 | + (−0.585 − 0.810i)2-s + 0.301·3-s + (−0.315 + 0.949i)4-s + 1.16i·5-s + (−0.176 − 0.244i)6-s − 0.782i·7-s + (0.953 − 0.299i)8-s − 0.908·9-s + (0.947 − 0.683i)10-s + 0.998·11-s + (−0.0950 + 0.286i)12-s − 1.13i·13-s + (−0.634 + 0.458i)14-s + 0.352i·15-s + (−0.801 − 0.598i)16-s − 1.66·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.299i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.953 + 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{19}{2})\) |
\(\approx\) |
\(0.0855194 - 0.557188i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0855194 - 0.557188i\) |
\(L(10)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (299. + 415. i)T \) |
good | 3 | \( 1 - 5.93e3T + 3.87e8T^{2} \) |
| 5 | \( 1 - 2.28e6iT - 3.81e12T^{2} \) |
| 7 | \( 1 + 3.15e7iT - 1.62e15T^{2} \) |
| 11 | \( 1 - 2.35e9T + 5.55e18T^{2} \) |
| 13 | \( 1 + 1.20e10iT - 1.12e20T^{2} \) |
| 17 | \( 1 + 1.97e11T + 1.40e22T^{2} \) |
| 19 | \( 1 + 1.48e11T + 1.04e23T^{2} \) |
| 23 | \( 1 + 1.21e12iT - 3.24e24T^{2} \) |
| 29 | \( 1 - 1.97e12iT - 2.10e26T^{2} \) |
| 31 | \( 1 + 2.27e13iT - 6.99e26T^{2} \) |
| 37 | \( 1 + 1.66e14iT - 1.68e28T^{2} \) |
| 41 | \( 1 + 2.66e14T + 1.07e29T^{2} \) |
| 43 | \( 1 + 7.20e14T + 2.52e29T^{2} \) |
| 47 | \( 1 + 1.62e15iT - 1.25e30T^{2} \) |
| 53 | \( 1 + 2.17e15iT - 1.08e31T^{2} \) |
| 59 | \( 1 - 1.10e16T + 7.50e31T^{2} \) |
| 61 | \( 1 - 6.26e15iT - 1.36e32T^{2} \) |
| 67 | \( 1 + 4.55e16T + 7.40e32T^{2} \) |
| 71 | \( 1 - 5.40e16iT - 2.10e33T^{2} \) |
| 73 | \( 1 - 4.71e14T + 3.46e33T^{2} \) |
| 79 | \( 1 - 1.72e17iT - 1.43e34T^{2} \) |
| 83 | \( 1 + 8.17e16T + 3.49e34T^{2} \) |
| 89 | \( 1 - 4.18e17T + 1.22e35T^{2} \) |
| 97 | \( 1 + 5.20e17T + 5.77e35T^{2} \) |
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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
−17.02128297819675251374726115574, −14.79777870184772321830758278526, −13.39764869412490313351905711893, −11.38981248683654252681420650068, −10.38367203682474591506143102799, −8.601249231017534116839385570492, −6.89126227282087735213999379511, −3.75987322373215949087351032185, −2.40721883812897041581624171666, −0.25080766117904725723032557294,
1.67770585309381920971517830821, 4.72014807740664297866971401199, 6.39158394613998981138378814671, 8.660185979669794930667128608018, 9.101383884462017101664631829406, 11.65096474313561704199596944798, 13.65369739025046581402907972963, 15.09681091832816807473635431924, 16.50689774136592223664900278339, 17.50192342140790337167486923256