| L(s) = 1 | + (3 − 5.19i)5-s + (18 + 31.1i)11-s + 62·13-s + (57 + 98.7i)17-s + (38 − 65.8i)19-s + (−12 + 20.7i)23-s + (44.5 + 77.0i)25-s − 54·29-s + (56 + 96.9i)31-s + (89 − 154. i)37-s − 378·41-s − 172·43-s + (−96 + 166. i)47-s + (−201 − 348. i)53-s + 216·55-s + ⋯ |
| L(s) = 1 | + (0.268 − 0.464i)5-s + (0.493 + 0.854i)11-s + 1.32·13-s + (0.813 + 1.40i)17-s + (0.458 − 0.794i)19-s + (−0.108 + 0.188i)23-s + (0.355 + 0.616i)25-s − 0.345·29-s + (0.324 + 0.561i)31-s + (0.395 − 0.684i)37-s − 1.43·41-s − 0.609·43-s + (−0.297 + 0.516i)47-s + (−0.520 − 0.902i)53-s + 0.529·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.537984225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.537984225\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (-3 + 5.19i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-18 - 31.1i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 62T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-57 - 98.7i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-38 + 65.8i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (12 - 20.7i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 54T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-56 - 96.9i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-89 + 154. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 378T + 6.89e4T^{2} \) |
| 43 | \( 1 + 172T + 7.95e4T^{2} \) |
| 47 | \( 1 + (96 - 166. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (201 + 348. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-198 - 342. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (127 - 219. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-506 - 876. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 840T + 3.57e5T^{2} \) |
| 73 | \( 1 + (445 + 770. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (40 - 69.2i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 108T + 5.71e5T^{2} \) |
| 89 | \( 1 + (819 - 1.41e3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.01e3T + 9.12e5T^{2} \) |
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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
−8.949089067676813534953711212213, −8.441200011104557433124453276930, −7.47029990351427067753005347970, −6.61889124880372168107656230822, −5.82410703443406472023309587185, −5.01606429151151892555810103087, −4.02088769281063797096125660513, −3.24262779659524419082455593518, −1.75236900299777579815509179670, −1.12547451513669133892225638442,
0.59878550989175080128156856499, 1.61695183772888529540617440184, 3.02855194034514112733958211792, 3.53044786399619899466539462131, 4.74165644082363399631367881562, 5.78773001286347016533481159173, 6.30815119834359154151319725987, 7.17816029611407992122616799436, 8.152136421632625136733224169077, 8.721452343943278549383145304237
