| L(s) = 1 | + (−15.4 − 44.1i)3-s + 125i·5-s − 3.33i·7-s + (−1.71e3 + 1.36e3i)9-s − 833.·11-s + 1.22e4·13-s + (5.51e3 − 1.92e3i)15-s − 1.20e4i·17-s − 3.83e4i·19-s + (−147. + 51.4i)21-s − 1.07e5·23-s − 1.56e4·25-s + (8.65e4 + 5.44e4i)27-s + 2.02e5i·29-s − 3.65e4i·31-s + ⋯ |
| L(s) = 1 | + (−0.330 − 0.943i)3-s + 0.447i·5-s − 0.00367i·7-s + (−0.782 + 0.623i)9-s − 0.188·11-s + 1.54·13-s + (0.422 − 0.147i)15-s − 0.592i·17-s − 1.28i·19-s + (−0.00346 + 0.00121i)21-s − 1.84·23-s − 0.199·25-s + (0.846 + 0.532i)27-s + 1.54i·29-s − 0.220i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.186 - 0.982i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.8268748818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8268748818\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (15.4 + 44.1i)T \) |
| 5 | \( 1 - 125iT \) |
| good | 7 | \( 1 + 3.33iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 833.T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.22e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.20e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 3.83e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 1.07e5T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.02e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 3.65e4iT - 2.75e10T^{2} \) |
| 37 | \( 1 + 3.38e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.13e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 + 6.19e4iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 9.58e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.57e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 1.95e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.08e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 8.33e5iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 2.51e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.38e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.63e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 - 1.20e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.27e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 - 1.27e7T + 8.07e13T^{2} \) |
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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
−11.21249895730605933746388869992, −10.38502527133379795050820541609, −8.944545590624697349171498080904, −8.056073751512720466592788160265, −6.97953821321695077082210885762, −6.24157195655739024356887408091, −5.15163830592915466837612061349, −3.53418035884994182907204649710, −2.27958748893752892443783836821, −1.06048637444739748311345784526,
0.22459794957348760514332704490, 1.73813064443906384602850779939, 3.57608390618044859100728184057, 4.22313480415061844759790109959, 5.65554497043317370420790592134, 6.19594867936565473254659271645, 8.050760303146312798480463000711, 8.687303146423792273698868734844, 9.900468515481859755572617096895, 10.50575256440786627547740773015
