| L(s) = 1 | − 8.27i·2-s + (−70.4 − 39.9i)3-s + 187.·4-s + (−330. + 582. i)6-s − 860.·7-s − 3.66e3i·8-s + (3.36e3 + 5.63e3i)9-s + 8.66e3i·11-s + (−1.32e4 − 7.50e3i)12-s + 1.84e4·13-s + 7.11e3i·14-s + 1.76e4·16-s + 1.10e5i·17-s + (4.65e4 − 2.78e4i)18-s − 7.12e4·19-s + ⋯ |
| L(s) = 1 | − 0.516i·2-s + (−0.869 − 0.493i)3-s + 0.732·4-s + (−0.255 + 0.449i)6-s − 0.358·7-s − 0.895i·8-s + (0.512 + 0.858i)9-s + 0.591i·11-s + (−0.637 − 0.361i)12-s + 0.646·13-s + 0.185i·14-s + 0.269·16-s + 1.31i·17-s + (0.443 − 0.264i)18-s − 0.546·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 + 0.493i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.869 + 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.68988 - 0.446207i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.68988 - 0.446207i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (70.4 + 39.9i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 8.27iT - 256T^{2} \) |
| 7 | \( 1 + 860.T + 5.76e6T^{2} \) |
| 11 | \( 1 - 8.66e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 1.84e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 1.10e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 7.12e4T + 1.69e10T^{2} \) |
| 23 | \( 1 - 3.69e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 3.06e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 1.39e6T + 8.52e11T^{2} \) |
| 37 | \( 1 - 3.68e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 5.05e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 1.42e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 7.79e5iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 7.55e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 2.08e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 2.03e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 1.37e6T + 4.06e14T^{2} \) |
| 71 | \( 1 + 4.19e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.57e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 1.07e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 1.63e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 5.98e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.07e8T + 7.83e15T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.60094216952020635706327867302, −11.67410424461360667233143975461, −10.77932889182616309570518277803, −9.858243634463250875840873298252, −7.940696531528284084469128678980, −6.70930372305419976522280372502, −5.86008020769868953890428515394, −3.98822616583802655798215740584, −2.20133486309970183598442499603, −1.01768345865244676290057678225,
0.76552306438986540600854172947, 2.89016179934099031533964660173, 4.66858633688787125545212342345, 6.05607672725789181855356215673, 6.69983238498707366778519756138, 8.251618219662707587213756199979, 9.737502513485741386760434074952, 10.95679477532996367881171241448, 11.59243047413018308044688671876, 12.82932459541600362350130682200
