L(s) = 1 | + (0.989 + 0.142i)2-s + (0.281 − 0.959i)3-s + (0.959 + 0.281i)4-s + (−0.540 − 0.841i)5-s + (0.415 − 0.909i)6-s + (0.909 + 0.415i)8-s + (−0.841 − 0.540i)9-s + (−0.415 − 0.909i)10-s + (0.540 − 0.841i)12-s + (−0.959 + 0.281i)15-s + (0.841 + 0.540i)16-s + (−0.989 − 0.857i)17-s + (−0.755 − 0.654i)18-s + (0.186 + 0.215i)19-s + (−0.281 − 0.959i)20-s + ⋯ |
L(s) = 1 | + (0.989 + 0.142i)2-s + (0.281 − 0.959i)3-s + (0.959 + 0.281i)4-s + (−0.540 − 0.841i)5-s + (0.415 − 0.909i)6-s + (0.909 + 0.415i)8-s + (−0.841 − 0.540i)9-s + (−0.415 − 0.909i)10-s + (0.540 − 0.841i)12-s + (−0.959 + 0.281i)15-s + (0.841 + 0.540i)16-s + (−0.989 − 0.857i)17-s + (−0.755 − 0.654i)18-s + (0.186 + 0.215i)19-s + (−0.281 − 0.959i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.403 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.403 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.952918748\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.952918748\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.989 - 0.142i)T \) |
| 3 | \( 1 + (-0.281 + 0.959i)T \) |
| 5 | \( 1 + (0.540 + 0.841i)T \) |
| 23 | \( 1 + (-0.909 + 0.415i)T \) |
good | 7 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 11 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 13 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 17 | \( 1 + (0.989 + 0.857i)T + (0.142 + 0.989i)T^{2} \) |
| 19 | \( 1 + (-0.186 - 0.215i)T + (-0.142 + 0.989i)T^{2} \) |
| 29 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 31 | \( 1 + (-0.557 - 1.89i)T + (-0.841 + 0.540i)T^{2} \) |
| 37 | \( 1 + (0.415 + 0.909i)T^{2} \) |
| 41 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 43 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 47 | \( 1 - 0.830iT - T^{2} \) |
| 53 | \( 1 + (-0.822 - 0.118i)T + (0.959 + 0.281i)T^{2} \) |
| 59 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
| 61 | \( 1 + (0.512 + 1.74i)T + (-0.841 + 0.540i)T^{2} \) |
| 67 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 71 | \( 1 + (-0.654 - 0.755i)T^{2} \) |
| 73 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 79 | \( 1 + (-0.239 - 1.66i)T + (-0.959 + 0.281i)T^{2} \) |
| 83 | \( 1 + (-0.474 - 0.304i)T + (0.415 + 0.909i)T^{2} \) |
| 89 | \( 1 + (0.841 + 0.540i)T^{2} \) |
| 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
−9.369900345082452541799251653529, −8.563042809421856102121183236276, −7.903779545037192575651305247595, −7.01751737889818520577796100569, −6.50156151593461874942843213453, −5.29604841475368165602305303640, −4.69279655854435554776725499589, −3.52076365169014484282905946206, −2.59606979862661149265299690383, −1.30848551940982463972500533558,
2.25566987190041024852053203383, 3.11430215489300614301822085638, 3.97439739436826483705263048355, 4.57571429529819229330426077792, 5.67448960376001204185061751612, 6.47289996189798824062539316180, 7.39912024776371993676567998696, 8.219815253924897567320308193308, 9.276332269329475282040364019877, 10.21581852423989760853176982584