L(s) = 1 | + (10.0 + 10.0i)2-s + (−112. + 84.4i)3-s − 311. i·4-s + (−1.96e3 − 277. i)6-s + (−2.70e3 + 2.70e3i)7-s + (8.24e3 − 8.24e3i)8-s + (5.43e3 − 1.89e4i)9-s − 2.51e4i·11-s + (2.62e4 + 3.48e4i)12-s + (1.31e5 + 1.31e5i)13-s − 5.42e4·14-s + 6.08e3·16-s + (−2.74e5 − 2.74e5i)17-s + (2.44e5 − 1.35e5i)18-s + 5.04e5i·19-s + ⋯ |
L(s) = 1 | + (0.442 + 0.442i)2-s + (−0.798 + 0.601i)3-s − 0.607i·4-s + (−0.620 − 0.0872i)6-s + (−0.426 + 0.426i)7-s + (0.712 − 0.712i)8-s + (0.275 − 0.961i)9-s − 0.518i·11-s + (0.365 + 0.485i)12-s + (1.28 + 1.28i)13-s − 0.377·14-s + 0.0231·16-s + (−0.796 − 0.796i)17-s + (0.547 − 0.303i)18-s + 0.887i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 - 0.363i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.931 - 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(0.157252 + 0.836605i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.157252 + 0.836605i\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (112. - 84.4i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-10.0 - 10.0i)T + 512iT^{2} \) |
| 7 | \( 1 + (2.70e3 - 2.70e3i)T - 4.03e7iT^{2} \) |
| 11 | \( 1 + 2.51e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 + (-1.31e5 - 1.31e5i)T + 1.06e10iT^{2} \) |
| 17 | \( 1 + (2.74e5 + 2.74e5i)T + 1.18e11iT^{2} \) |
| 19 | \( 1 - 5.04e5iT - 3.22e11T^{2} \) |
| 23 | \( 1 + (3.57e5 - 3.57e5i)T - 1.80e12iT^{2} \) |
| 29 | \( 1 - 1.44e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.57e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + (1.41e7 - 1.41e7i)T - 1.29e14iT^{2} \) |
| 41 | \( 1 - 1.52e7iT - 3.27e14T^{2} \) |
| 43 | \( 1 + (1.45e7 + 1.45e7i)T + 5.02e14iT^{2} \) |
| 47 | \( 1 + (2.49e7 + 2.49e7i)T + 1.11e15iT^{2} \) |
| 53 | \( 1 + (2.38e7 - 2.38e7i)T - 3.29e15iT^{2} \) |
| 59 | \( 1 + 1.41e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 2.77e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + (5.16e7 - 5.16e7i)T - 2.72e16iT^{2} \) |
| 71 | \( 1 - 2.50e8iT - 4.58e16T^{2} \) |
| 73 | \( 1 + (-8.81e7 - 8.81e7i)T + 5.88e16iT^{2} \) |
| 79 | \( 1 - 3.89e8iT - 1.19e17T^{2} \) |
| 83 | \( 1 + (9.47e6 - 9.47e6i)T - 1.86e17iT^{2} \) |
| 89 | \( 1 - 4.88e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + (7.00e7 - 7.00e7i)T - 7.60e17iT^{2} \) |
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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
−13.37031109354449465528453526820, −11.88113617588189535823957147305, −10.99647811777532838501686490836, −9.874370911779607071303223082583, −8.852743901908296818185789530131, −6.68888079620289409577796391295, −6.07513551814724745151501941493, −4.90161487111987670670090269583, −3.70472018611425232347737424191, −1.36658689287492063801879381930,
0.24161129945717239018415563673, 1.84634678207708784575610327993, 3.41475611099982106522026128923, 4.75864856313112164984674773299, 6.21789660621840296412034067917, 7.39538845335171503527473728902, 8.529827941878913382244329087414, 10.51656995976314725805970637949, 11.12871681791001268678371336323, 12.43072408595513262562423427388