| L(s) = 1 | + (−27.9 − 37.5i)3-s − 125i·5-s + 1.23e3i·7-s + (−628. + 2.09e3i)9-s − 4.91e3·11-s − 7.45e3·13-s + (−4.68e3 + 3.48e3i)15-s + 1.92e4i·17-s − 2.59e4i·19-s + (4.62e4 − 3.44e4i)21-s − 3.33e4·23-s − 1.56e4·25-s + (9.61e4 − 3.49e4i)27-s + 8.13e3i·29-s + 2.00e5i·31-s + ⋯ |
| L(s) = 1 | + (−0.596 − 0.802i)3-s − 0.447i·5-s + 1.35i·7-s + (−0.287 + 0.957i)9-s − 1.11·11-s − 0.941·13-s + (−0.358 + 0.266i)15-s + 0.949i·17-s − 0.867i·19-s + (1.09 − 0.811i)21-s − 0.572·23-s − 0.199·25-s + (0.939 − 0.341i)27-s + 0.0619i·29-s + 1.20i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.115 + 0.993i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.115 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.7687060970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7687060970\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (27.9 + 37.5i)T \) |
| 5 | \( 1 + 125iT \) |
| good | 7 | \( 1 - 1.23e3iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 4.91e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 7.45e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.92e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 2.59e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 3.33e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 8.13e3iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 2.00e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 - 4.91e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 7.75e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 - 9.88e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 5.90e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.50e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 1.08e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.41e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.00e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 2.32e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.70e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.74e3iT - 1.92e13T^{2} \) |
| 83 | \( 1 + 7.56e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.08e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 + 1.45e5T + 8.07e13T^{2} \) |
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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
−10.84744247238505965595465050729, −9.670288349909910340445934148953, −8.489677012033101993056008635811, −7.78179919868960631990100512159, −6.50638789200227374603559476290, −5.51060871828514936647818216103, −4.84026782981880964226633410968, −2.72614506018246209715855224685, −1.87210814594785999404838488680, −0.29718273090387220190604449493,
0.67567222421103645059693274752, 2.64859517829863484289212316043, 3.91731451564268857038310194081, 4.81451507219215283641177842670, 5.94067566972625134237699127268, 7.16842333407367723538036943247, 7.944994101440147221188236685025, 9.712878158420956583767518271621, 10.08223323535802649606618813117, 10.96235504353082041528754378377
