| L(s) = 1 | + (−2.82 − 2.82i)2-s + 16.0i·4-s + (117. − 117. i)7-s + (45.2 − 45.2i)8-s + 28.2i·11-s + (283. + 283. i)13-s − 662.·14-s − 256.·16-s + (137. + 137. i)17-s − 2.80e3i·19-s + (79.9 − 79.9i)22-s + (902. − 902. i)23-s − 1.60e3i·26-s + (1.87e3 + 1.87e3i)28-s + 827.·29-s + ⋯ |
| L(s) = 1 | + (−0.499 − 0.499i)2-s + 0.500i·4-s + (0.903 − 0.903i)7-s + (0.250 − 0.250i)8-s + 0.0704i·11-s + (0.464 + 0.464i)13-s − 0.903·14-s − 0.250·16-s + (0.115 + 0.115i)17-s − 1.78i·19-s + (0.0352 − 0.0352i)22-s + (0.355 − 0.355i)23-s − 0.464i·26-s + (0.451 + 0.451i)28-s + 0.182·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.279 + 0.960i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.714477228\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.714477228\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.82 + 2.82i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-117. + 117. i)T - 1.68e4iT^{2} \) |
| 11 | \( 1 - 28.2iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (-283. - 283. i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + (-137. - 137. i)T + 1.41e6iT^{2} \) |
| 19 | \( 1 + 2.80e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + (-902. + 902. i)T - 6.43e6iT^{2} \) |
| 29 | \( 1 - 827.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.04e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (1.32e3 - 1.32e3i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 - 1.07e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (7.83e3 + 7.83e3i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + (-1.17e4 - 1.17e4i)T + 2.29e8iT^{2} \) |
| 53 | \( 1 + (2.25e4 - 2.25e4i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 - 3.31e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.20e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (-2.74e4 + 2.74e4i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 + 5.46e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-1.42e4 - 1.42e4i)T + 2.07e9iT^{2} \) |
| 79 | \( 1 + 7.92e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + (3.65e4 - 3.65e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 + 6.99e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-6.93e4 + 6.93e4i)T - 8.58e9iT^{2} \) |
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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
−10.12455392396226948332955768923, −9.118013987713293100799775170934, −8.320329265699145412429016679238, −7.38469212407931296926275932089, −6.55570793202758105772803591173, −4.92980061812472813266174801744, −4.16282679644567458150021715017, −2.83355238049205813623884089840, −1.53195783224916479523811977081, −0.55969050989270357018263084560,
1.10583533568007124372199529901, 2.20069722012447490281315623999, 3.73982976927084696774662174271, 5.21707575041145699666063048689, 5.75758668898654898385838750901, 6.93349530994194759734012290552, 8.188490392704496504487181087834, 8.379679101610344971071899683766, 9.580454274916697417311728986186, 10.42009706853985421605851606053
