L(s) = 1 | + (−2 + 3.46i)5-s + (−2.5 + 0.866i)7-s + (3 + 5.19i)11-s + 5·13-s + (1 + 1.73i)17-s + (−0.5 + 0.866i)19-s + (−3 + 5.19i)23-s + (−5.49 − 9.52i)25-s + (1.5 + 2.59i)31-s + (2.00 − 10.3i)35-s + (−1.5 + 2.59i)37-s + 6·41-s + 5·43-s + (−2 + 3.46i)47-s + (5.5 − 4.33i)49-s + ⋯ |
L(s) = 1 | + (−0.894 + 1.54i)5-s + (−0.944 + 0.327i)7-s + (0.904 + 1.56i)11-s + 1.38·13-s + (0.242 + 0.420i)17-s + (−0.114 + 0.198i)19-s + (−0.625 + 1.08i)23-s + (−1.09 − 1.90i)25-s + (0.269 + 0.466i)31-s + (0.338 − 1.75i)35-s + (−0.246 + 0.427i)37-s + 0.937·41-s + 0.762·43-s + (−0.291 + 0.505i)47-s + (0.785 − 0.618i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.083012667\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.083012667\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.5 - 0.866i)T \) |
good | 5 | \( 1 + (2 - 3.46i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (-1.5 - 2.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 + (2 - 3.46i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.5 + 6.06i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 + (-1.5 - 2.59i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + (-2 + 3.46i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−9.705669078799358469279935259753, −8.748867424409151745968061019412, −7.76732161916545072025508817617, −7.12186113737288095602389529301, −6.44986139641714886459493366663, −5.91783845579342137278276328283, −4.29361359603048031755335932096, −3.68102080081483078686252494703, −2.98474324475393554791208926417, −1.71464561998985268449813331184,
0.45630287248820541482970382664, 1.15789215053590767730341613715, 3.09834762131747131662728026251, 3.93509239316057844557849880793, 4.40511545942400333726023419678, 5.81357087601946103823510898100, 6.14073094259087855165751058282, 7.32443263578210754350423006043, 8.228226317465367332758961560617, 8.874871154837142022026726549725