L(s) = 1 | + (0.5 − 0.866i)3-s − 5-s + (−2.5 − 4.33i)7-s + (1 + 1.73i)9-s + (2.5 − 4.33i)11-s + (−1 − 3.46i)13-s + (−0.5 + 0.866i)15-s + (0.5 + 0.866i)17-s + (1.5 + 2.59i)19-s − 5·21-s + (−1.5 + 2.59i)23-s + 25-s + 5·27-s + (0.5 − 0.866i)29-s + (−2.5 − 4.33i)33-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s − 0.447·5-s + (−0.944 − 1.63i)7-s + (0.333 + 0.577i)9-s + (0.753 − 1.30i)11-s + (−0.277 − 0.960i)13-s + (−0.129 + 0.223i)15-s + (0.121 + 0.210i)17-s + (0.344 + 0.596i)19-s − 1.09·21-s + (−0.312 + 0.541i)23-s + 0.200·25-s + 0.962·27-s + (0.0928 − 0.160i)29-s + (−0.435 − 0.753i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.822973 - 0.812487i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.822973 - 0.812487i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + (1 + 3.46i)T \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (2.5 + 4.33i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.5 - 2.59i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.5 + 4.33i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.5 + 4.33i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + (-5.5 - 9.52i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.5 - 2.59i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (6.5 + 11.2i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (3.5 - 6.06i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.5 + 9.52i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.82652505519072324707373203067, −10.59128653523405392252581592363, −10.13230188100752471625254207499, −8.673837985662491270895810324127, −7.65241651154255264766333961869, −7.04654188718715583010379475782, −5.81242186146166551164724814922, −4.08004395856611851986330547778, −3.21492978616317955927313025929, −0.938552208271945488268348878911,
2.37637429776689080483134120293, 3.74217712127875368808055793406, 4.85857126071765306658023207758, 6.34250498390718988551963612027, 7.12470978442532819385875394780, 8.773381552258027316656363259159, 9.341937425564374435374481987126, 9.931394610781556530995278621074, 11.56025846344221747983006624058, 12.24924731043351185670842179444