L(s) = 1 | + (0.654 + 0.755i)7-s + (−0.959 − 0.281i)9-s + (0.239 + 0.153i)11-s + (0.959 − 0.281i)23-s + (0.841 − 0.540i)25-s + (0.698 + 1.53i)29-s + (1.25 + 0.368i)37-s + (−0.273 + 1.89i)43-s + (−0.142 + 0.989i)49-s + (−1.10 − 1.27i)53-s + (−0.415 − 0.909i)63-s + (−0.698 + 0.449i)67-s + (0.239 − 0.153i)71-s + (0.0405 + 0.281i)77-s + (1.10 − 1.27i)79-s + ⋯ |
L(s) = 1 | + (0.654 + 0.755i)7-s + (−0.959 − 0.281i)9-s + (0.239 + 0.153i)11-s + (0.959 − 0.281i)23-s + (0.841 − 0.540i)25-s + (0.698 + 1.53i)29-s + (1.25 + 0.368i)37-s + (−0.273 + 1.89i)43-s + (−0.142 + 0.989i)49-s + (−1.10 − 1.27i)53-s + (−0.415 − 0.909i)63-s + (−0.698 + 0.449i)67-s + (0.239 − 0.153i)71-s + (0.0405 + 0.281i)77-s + (1.10 − 1.27i)79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.251555320\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.251555320\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.654 - 0.755i)T \) |
| 23 | \( 1 + (-0.959 + 0.281i)T \) |
good | 3 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 5 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 11 | \( 1 + (-0.239 - 0.153i)T + (0.415 + 0.909i)T^{2} \) |
| 13 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 17 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 19 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 29 | \( 1 + (-0.698 - 1.53i)T + (-0.654 + 0.755i)T^{2} \) |
| 31 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 37 | \( 1 + (-1.25 - 0.368i)T + (0.841 + 0.540i)T^{2} \) |
| 41 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 43 | \( 1 + (0.273 - 1.89i)T + (-0.959 - 0.281i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (1.10 + 1.27i)T + (-0.142 + 0.989i)T^{2} \) |
| 59 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 61 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 67 | \( 1 + (0.698 - 0.449i)T + (0.415 - 0.909i)T^{2} \) |
| 71 | \( 1 + (-0.239 + 0.153i)T + (0.415 - 0.909i)T^{2} \) |
| 73 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 79 | \( 1 + (-1.10 + 1.27i)T + (-0.142 - 0.989i)T^{2} \) |
| 83 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 89 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 97 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
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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
−8.982499839803303541410482937256, −8.484541517562006646989334735488, −7.79046451540367529480965718641, −6.67585711259028401431391325939, −6.13423265169856030265048764177, −5.09964473154496891267550592631, −4.64779563263651610540928335591, −3.23015965170973645336715208903, −2.61250824555255549824477473052, −1.29109435694375307851471245811,
0.972788070586442169679869966234, 2.32029123447014045477016489030, 3.29772351670634653311982787898, 4.29418596908107989941081453493, 5.06476033387860507175572275229, 5.87031118072455753810867236447, 6.76823755292522251807927365088, 7.57931014183801098138520031533, 8.210935964156892185003404720681, 8.936164468507284085106944261538