L(s) = 1 | + (−1.29 + 1.15i)3-s + (0.555 − 0.962i)5-s + (0.5 + 0.866i)7-s + (0.349 − 2.97i)9-s + (−0.944 − 1.63i)11-s + (0.5 − 0.866i)13-s + (0.388 + 1.88i)15-s − 5.87·17-s − 7.09·19-s + (−1.64 − 0.545i)21-s + (1.99 − 3.45i)23-s + (1.88 + 3.26i)25-s + (2.97 + 4.25i)27-s + (−0.493 − 0.855i)29-s + (−0.333 + 0.576i)31-s + ⋯ |
L(s) = 1 | + (−0.747 + 0.664i)3-s + (0.248 − 0.430i)5-s + (0.188 + 0.327i)7-s + (0.116 − 0.993i)9-s + (−0.284 − 0.493i)11-s + (0.138 − 0.240i)13-s + (0.100 + 0.486i)15-s − 1.42·17-s − 1.62·19-s + (−0.358 − 0.118i)21-s + (0.415 − 0.720i)23-s + (0.376 + 0.652i)25-s + (0.572 + 0.819i)27-s + (−0.0916 − 0.158i)29-s + (−0.0598 + 0.103i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.449 + 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.449 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4837298755\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4837298755\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.29 - 1.15i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-0.555 + 0.962i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.944 + 1.63i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 5.87T + 17T^{2} \) |
| 19 | \( 1 + 7.09T + 19T^{2} \) |
| 23 | \( 1 + (-1.99 + 3.45i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.493 + 0.855i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.333 - 0.576i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 1.33T + 37T^{2} \) |
| 41 | \( 1 + (-0.944 + 1.63i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.43 + 9.40i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (5.54 + 9.61i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + (-2.38 + 4.12i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.88 + 3.26i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.04 + 3.54i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 3.09T + 73T^{2} \) |
| 79 | \( 1 + (-3.21 - 5.56i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.93 + 10.2i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 + (0.382 + 0.662i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.727301196865891550656652575601, −8.795439945269268940027072811206, −8.436069639575369457496223817290, −6.85065827159258799891142850631, −6.22871939482648724729094109721, −5.21908306912602478278134954361, −4.62160731199952458491714248749, −3.51781465126373213347651170676, −2.04913516522064541517931265466, −0.23217251909903247647741647832,
1.62167878771215242873806777183, 2.61866684690192721676212619497, 4.30130923997797251535066553645, 4.98727529227847827028519649379, 6.39211734609099916343107347584, 6.55368203662525336342104224594, 7.61232945483733032453003678746, 8.416454701522404598320322538511, 9.493502844715270540716100365584, 10.52906550947625689016200270913