L(s) = 1 | + 55.9·5-s − 245. i·7-s − 243·9-s + 802i·11-s − 245.·13-s + 1.91e3i·19-s + 3.33e3i·23-s + 3.12e3·25-s − 1.37e4i·35-s + 1.58e4·37-s + 1.52e4·41-s − 1.35e4·45-s − 1.53e4i·47-s − 4.36e4·49-s + 1.68e4·53-s + ⋯ |
L(s) = 1 | + 0.999·5-s − 1.89i·7-s − 9-s + 1.99i·11-s − 0.403·13-s + 1.21i·19-s + 1.31i·23-s + 25-s − 1.89i·35-s + 1.90·37-s + 1.41·41-s − 0.999·45-s − 1.01i·47-s − 2.59·49-s + 0.823·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.006085520\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.006085520\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - 55.9T \) |
good | 3 | \( 1 + 243T^{2} \) |
| 7 | \( 1 + 245. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 802iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 245.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.91e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 3.33e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 2.05e7T^{2} \) |
| 31 | \( 1 + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.58e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.52e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.53e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.68e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.79e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 8.44e8T^{2} \) |
| 67 | \( 1 + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.07e9T^{2} \) |
| 79 | \( 1 + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.28e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.58e9T^{2} \) |
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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
−10.65656454220006810757423457411, −9.978587801282841996252861914315, −9.399940886867519002305638119999, −7.77564199878126322771726769766, −7.19441764679067891009491935754, −6.05007141337034201631350484736, −4.86691942303921970454520533169, −3.84282787951566465870555092599, −2.27675507796219162943257077935, −1.11128989406410684344367236482,
0.57299733356626466473965749556, 2.52865423711880795406053004803, 2.82040731214383640035707393217, 5.01186322965047259224628202810, 5.92976261812794858455445699862, 6.23810048339743478371387697424, 8.226305163190499810521362711124, 8.883143342167209417097288441311, 9.407332922491448835665708566608, 10.93483592500462354177732095138