| L(s) = 1 | + (12.0 − 24.1i)3-s + 436. i·7-s + (−437. − 582. i)9-s − 1.10e3i·11-s + 908. i·13-s + 9.74e3·17-s − 7.70e3·19-s + (1.05e4 + 5.27e3i)21-s − 1.93e4·23-s + (−1.93e4 + 3.54e3i)27-s − 8.10e3i·29-s − 768.·31-s + (−2.65e4 − 1.32e4i)33-s − 7.22e4i·37-s + (2.19e4 + 1.09e4i)39-s + ⋯ |
| L(s) = 1 | + (0.446 − 0.894i)3-s + 1.27i·7-s + (−0.600 − 0.799i)9-s − 0.826i·11-s + 0.413i·13-s + 1.98·17-s − 1.12·19-s + (1.13 + 0.569i)21-s − 1.58·23-s + (−0.983 + 0.180i)27-s − 0.332i·29-s − 0.0258·31-s + (−0.739 − 0.369i)33-s − 1.42i·37-s + (0.369 + 0.184i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.000383i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.999 + 0.000383i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.7182350006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7182350006\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-12.0 + 24.1i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 436. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 1.10e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 908. iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 9.74e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 7.70e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 1.93e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + 8.10e3iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 768.T + 8.87e8T^{2} \) |
| 37 | \( 1 + 7.22e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 1.40e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 1.21e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 3.70e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + 1.73e5T + 2.21e10T^{2} \) |
| 59 | \( 1 - 1.38e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 2.78e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 9.62e4iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 3.38e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 3.75e4iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 7.21e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 3.10e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 1.11e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 3.06e5iT - 8.32e11T^{2} \) |
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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
−10.12997961934322726714342941197, −9.004745551214992728966335305571, −8.362423251275164538574863092389, −7.48781505931167059529145057554, −6.09859202275949495769064413888, −5.66480368919695932334476826779, −3.77167517417396912204357986370, −2.62707316362326504464427613233, −1.64260548942010289260429727566, −0.15246632321768931633454443680,
1.49826332247757035357538202107, 3.08745231787820786347022668738, 4.05065194481502218050153168000, 4.89938291922723916353864024658, 6.24053554987088808424407667362, 7.66625696191914842561815516580, 8.139445089196255485467483810034, 9.675528234202947490522961921871, 10.10097677748597832223536901644, 10.82994044957435585592345609500
