L(s) = 1 | + 1.26e5i·2-s − 8.63e7i·3-s − 7.38e9·4-s + 1.09e13·6-s − 7.78e13i·7-s + 1.52e14i·8-s − 1.89e15·9-s + 1.61e17·11-s + 6.37e17i·12-s − 2.28e18i·13-s + 9.84e18·14-s − 8.26e19·16-s − 2.46e20i·17-s − 2.39e20i·18-s + 1.53e21·19-s + ⋯ |
L(s) = 1 | + 1.36i·2-s − 1.15i·3-s − 0.859·4-s + 1.57·6-s − 0.885i·7-s + 0.190i·8-s − 0.340·9-s + 1.06·11-s + 0.995i·12-s − 0.953i·13-s + 1.20·14-s − 1.12·16-s − 1.22i·17-s − 0.464i·18-s + 1.22·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(17)\) |
\(\approx\) |
\(2.421271244\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.421271244\) |
\(L(\frac{35}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 - 1.26e5iT - 8.58e9T^{2} \) |
| 3 | \( 1 + 8.63e7iT - 5.55e15T^{2} \) |
| 7 | \( 1 + 7.78e13iT - 7.73e27T^{2} \) |
| 11 | \( 1 - 1.61e17T + 2.32e34T^{2} \) |
| 13 | \( 1 + 2.28e18iT - 5.75e36T^{2} \) |
| 17 | \( 1 + 2.46e20iT - 4.02e40T^{2} \) |
| 19 | \( 1 - 1.53e21T + 1.58e42T^{2} \) |
| 23 | \( 1 + 2.16e22iT - 8.65e44T^{2} \) |
| 29 | \( 1 - 2.07e23T + 1.81e48T^{2} \) |
| 31 | \( 1 - 1.75e24T + 1.64e49T^{2} \) |
| 37 | \( 1 + 8.12e25iT - 5.63e51T^{2} \) |
| 41 | \( 1 + 1.92e26T + 1.66e53T^{2} \) |
| 43 | \( 1 - 8.01e26iT - 8.02e53T^{2} \) |
| 47 | \( 1 - 7.53e27iT - 1.51e55T^{2} \) |
| 53 | \( 1 - 1.75e28iT - 7.96e56T^{2} \) |
| 59 | \( 1 - 2.97e29T + 2.74e58T^{2} \) |
| 61 | \( 1 + 3.70e29T + 8.23e58T^{2} \) |
| 67 | \( 1 + 8.43e29iT - 1.82e60T^{2} \) |
| 71 | \( 1 - 5.92e30T + 1.23e61T^{2} \) |
| 73 | \( 1 - 7.49e30iT - 3.08e61T^{2} \) |
| 79 | \( 1 + 6.07e30T + 4.18e62T^{2} \) |
| 83 | \( 1 - 3.47e31iT - 2.13e63T^{2} \) |
| 89 | \( 1 + 1.80e32T + 2.13e64T^{2} \) |
| 97 | \( 1 + 1.09e33iT - 3.65e65T^{2} \) |
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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
−11.27000181051433100829356760578, −9.583078690694664387238123455100, −8.161048427139709221348171252909, −7.31897477792134968848631340029, −6.78469073980243290122171354569, −5.69672952262498442218788803440, −4.40336730566162496818589336163, −2.76865270434688444327370234241, −1.22450504257602700691331416713, −0.50016107173652805996046967018,
1.19812121222543098061826626943, 2.06667259525978232056771891243, 3.43301629114237665417918929108, 3.98709648103495369937593367197, 5.19269208926660243135698163012, 6.68529454759073505808639205017, 8.765720903277811405221571352071, 9.525496509577793694579678921934, 10.31820355141290402889945157595, 11.57244629782409086430162114320