| L(s) = 1 | − 19.1·3-s + (55.2 − 8.66i)5-s − 121. i·7-s + 123.·9-s − 298i·11-s − 485.·13-s + (−1.05e3 + 165. i)15-s + 420. i·17-s − 2.57e3i·19-s + 2.32e3i·21-s − 717. i·23-s + (2.97e3 − 956. i)25-s + 2.29e3·27-s + 3.73e3i·29-s + 6.22e3·31-s + ⋯ |
| L(s) = 1 | − 1.22·3-s + (0.987 − 0.154i)5-s − 0.937i·7-s + 0.506·9-s − 0.742i·11-s − 0.797·13-s + (−1.21 + 0.190i)15-s + 0.353i·17-s − 1.63i·19-s + 1.15i·21-s − 0.282i·23-s + (0.952 − 0.306i)25-s + 0.606·27-s + 0.824i·29-s + 1.16·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.106i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5366413978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5366413978\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-55.2 + 8.66i)T \) |
| good | 3 | \( 1 + 19.1T + 243T^{2} \) |
| 7 | \( 1 + 121. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 298iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 485.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 420. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.57e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 717. iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 3.73e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 6.22e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.72e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.13e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.91e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.28e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 1.72e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.63e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 4.87e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 8.39e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.90e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 420. iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 3.11e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.06e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.93e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.80e5iT - 8.58e9T^{2} \) |
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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
−10.54348732682148822325023206317, −9.614231139222544214800991409476, −8.504693313299755175461550995563, −7.02437210273869480695895171902, −6.38061526438215291960934968734, −5.31776056834574124469597201183, −4.58483398711268216371129150238, −2.83314352951830629074152949656, −1.16831888350469217130965480641, −0.17910271212036626436635804063,
1.55158919650606585430643920607, 2.70465077125751782100970246009, 4.63627020526938312961526069356, 5.57298993660533495180672644757, 6.10824767473581513102015253841, 7.18167564476570847258834593362, 8.545736692859866687370256216162, 9.839256417750570459537711505991, 10.13611864452427091317818852721, 11.43549055057487243767374920476
