| L(s) = 1 | + (21.5 + 51.5i)5-s + 168.·7-s + 307.·11-s + 191. i·13-s − 1.13e3·17-s + 1.27e3i·19-s + 2.34e3i·23-s + (−2.19e3 + 2.22e3i)25-s − 1.05e3i·29-s + 3.22e3i·31-s + (3.64e3 + 8.70e3i)35-s + 1.19e4i·37-s − 1.04e4i·41-s − 4.14e3·43-s − 2.96e3i·47-s + ⋯ |
| L(s) = 1 | + (0.386 + 0.922i)5-s + 1.30·7-s + 0.766·11-s + 0.314i·13-s − 0.949·17-s + 0.809i·19-s + 0.925i·23-s + (−0.701 + 0.712i)25-s − 0.233i·29-s + 0.602i·31-s + (0.502 + 1.20i)35-s + 1.43i·37-s − 0.970i·41-s − 0.342·43-s − 0.196i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 - 0.883i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.469 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.438797698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.438797698\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-21.5 - 51.5i)T \) |
| good | 7 | \( 1 - 168.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 307.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 191. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.13e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.27e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 2.34e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 1.05e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 3.22e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 1.19e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.04e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 4.14e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.96e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 8.22e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.73e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.81e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.44e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.62e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.72e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 379. iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 3.06e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 1.12e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 8.43e4iT - 8.58e9T^{2} \) |
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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
−10.01891177892865745402113306873, −9.056969147038184159987493465254, −8.199109710136001546578293694301, −7.25706102496910158273620075639, −6.49071767628289388736348419491, −5.51090767182283938843677348379, −4.45881092575232149232699952920, −3.45478461995584590462439384095, −2.11621524698125830724514674524, −1.39263482159596347707984021047,
0.48559467195765507609712106917, 1.49499263687036784562665787995, 2.43311276568202052539529289319, 4.16169943764312434447542555128, 4.74212105731410932938110910034, 5.65917531937135372262960698887, 6.70542888748546992825129682562, 7.79193500693344048171437576497, 8.669306999008386533702811122991, 9.100091928956644818655749385859
