L(s) = 1 | + (−2.5 + 4.33i)5-s + (1.55 + 2.68i)7-s + (20.2 + 35.0i)11-s + (16.7 − 28.9i)13-s + 70.5·17-s + 146.·19-s + (−50.2 + 87.0i)23-s + (−12.5 − 21.6i)25-s + (−50.2 − 87.0i)29-s + (12.9 − 22.4i)31-s − 15.5·35-s − 218.·37-s + (46.5 − 80.6i)41-s + (−8.63 − 14.9i)43-s + (−206. − 357. i)47-s + ⋯ |
L(s) = 1 | + (−0.223 + 0.387i)5-s + (0.0838 + 0.145i)7-s + (0.555 + 0.961i)11-s + (0.356 − 0.617i)13-s + 1.00·17-s + 1.76·19-s + (−0.455 + 0.789i)23-s + (−0.100 − 0.173i)25-s + (−0.321 − 0.557i)29-s + (0.0750 − 0.129i)31-s − 0.0749·35-s − 0.971·37-s + (0.177 − 0.307i)41-s + (−0.0306 − 0.0530i)43-s + (−0.640 − 1.10i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.388749543\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.388749543\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.5 - 4.33i)T \) |
good | 7 | \( 1 + (-1.55 - 2.68i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-20.2 - 35.0i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-16.7 + 28.9i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 - 70.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 146.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (50.2 - 87.0i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (50.2 + 87.0i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-12.9 + 22.4i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 218.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-46.5 + 80.6i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (8.63 + 14.9i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (206. + 357. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 471.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (321. - 556. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (245. + 424. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (218. - 378. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 1.11e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 222.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-201. - 349. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-330. - 571. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 - 173.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-295. - 510. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−9.269590656471205883362266035925, −8.171447191977155380544876871155, −7.50277791069195186123831595254, −6.88890117694245774738726633997, −5.73007749789249656324601020149, −5.16668772822770495107353162723, −3.87375774338204350875735033111, −3.26377679197488960829707892714, −1.99559088519357962679497357648, −0.881368529721790449658195195640,
0.71081366311811625446271859026, 1.52973220953015020772436594560, 3.09957054149778554872326080061, 3.76083967485491057766468119154, 4.83099373519599163331704781349, 5.67104664148513234714497972993, 6.47715581278334605242972451699, 7.46291977429088075249079750515, 8.137602189783427080495199235753, 8.998647008719550503136079351829