L(s) = 1 | + (−0.415 + 0.909i)2-s + (0.959 + 0.281i)3-s + (−0.654 − 0.755i)4-s + (−1.10 + 0.708i)5-s + (−0.654 + 0.755i)6-s + (0.142 − 0.989i)7-s + (0.959 − 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.186 − 1.29i)10-s + (0.118 + 0.258i)11-s + (−0.415 − 0.909i)12-s + (0.841 + 0.540i)14-s + (−1.25 + 0.368i)15-s + (−0.142 + 0.989i)16-s + (1.25 − 1.45i)17-s + (−0.841 + 0.540i)18-s + ⋯ |
L(s) = 1 | + (−0.415 + 0.909i)2-s + (0.959 + 0.281i)3-s + (−0.654 − 0.755i)4-s + (−1.10 + 0.708i)5-s + (−0.654 + 0.755i)6-s + (0.142 − 0.989i)7-s + (0.959 − 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.186 − 1.29i)10-s + (0.118 + 0.258i)11-s + (−0.415 − 0.909i)12-s + (0.841 + 0.540i)14-s + (−1.25 + 0.368i)15-s + (−0.142 + 0.989i)16-s + (1.25 − 1.45i)17-s + (−0.841 + 0.540i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.101553521\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.101553521\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.415 - 0.909i)T \) |
| 3 | \( 1 + (-0.959 - 0.281i)T \) |
| 7 | \( 1 + (-0.142 + 0.989i)T \) |
| 23 | \( 1 + (-0.654 - 0.755i)T \) |
good | 5 | \( 1 + (1.10 - 0.708i)T + (0.415 - 0.909i)T^{2} \) |
| 11 | \( 1 + (-0.118 - 0.258i)T + (-0.654 + 0.755i)T^{2} \) |
| 13 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 17 | \( 1 + (-1.25 + 1.45i)T + (-0.142 - 0.989i)T^{2} \) |
| 19 | \( 1 + (-0.544 - 0.627i)T + (-0.142 + 0.989i)T^{2} \) |
| 29 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 31 | \( 1 + (1.84 - 0.540i)T + (0.841 - 0.540i)T^{2} \) |
| 37 | \( 1 + (-1.41 - 0.909i)T + (0.415 + 0.909i)T^{2} \) |
| 41 | \( 1 + (-0.698 + 0.449i)T + (0.415 - 0.909i)T^{2} \) |
| 43 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 59 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 61 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 67 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 71 | \( 1 + (0.698 - 1.53i)T + (-0.654 - 0.755i)T^{2} \) |
| 73 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 79 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 83 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 89 | \( 1 + (1.61 + 0.474i)T + (0.841 + 0.540i)T^{2} \) |
| 97 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
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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
−9.625600984677233484195128522514, −8.603937664833349603545303117616, −7.68354915376130400713178931783, −7.39992930125645784462247703021, −7.03560989690156244486502836320, −5.54822871469440981992458599621, −4.60844308745719930733223652008, −3.77589839455902142275590275895, −3.10300630184352157547617045827, −1.27393499840124776103910969104,
1.07183088674738576393901067799, 2.23279470702427245232687275797, 3.25437790820295314352774326982, 3.91926166553177439953054990777, 4.79669577202943415696159791936, 5.95567800540139755381266163834, 7.46837516924007667883297718096, 7.86035198237826180037402047913, 8.599751796083228068423346892288, 9.038070482244278524805942294745