L(s) = 1 | + (0.513 + 2.78i)2-s + (−7.47 + 2.85i)4-s − 2.40i·5-s − 2.65i·7-s + (−11.7 − 19.3i)8-s + (6.70 − 1.23i)10-s + 48.2·11-s + 40.7·13-s + (7.39 − 1.36i)14-s + (47.6 − 42.6i)16-s + 36.3i·17-s + 125. i·19-s + (6.87 + 18.0i)20-s + (24.7 + 134. i)22-s − 194.·23-s + ⋯ |
L(s) = 1 | + (0.181 + 0.983i)2-s + (−0.934 + 0.356i)4-s − 0.215i·5-s − 0.143i·7-s + (−0.520 − 0.853i)8-s + (0.211 − 0.0391i)10-s + 1.32·11-s + 0.868·13-s + (0.141 − 0.0260i)14-s + (0.745 − 0.666i)16-s + 0.518i·17-s + 1.51i·19-s + (0.0769 + 0.201i)20-s + (0.240 + 1.30i)22-s − 1.75·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.845842020\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.845842020\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.513 - 2.78i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 2.40iT - 125T^{2} \) |
| 7 | \( 1 + 2.65iT - 343T^{2} \) |
| 11 | \( 1 - 48.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 40.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 36.3iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 125. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 194.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 177. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 175. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 199.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 233. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 291. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 61.8T + 1.03e5T^{2} \) |
| 53 | \( 1 - 352. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 141.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 15.4T + 2.26e5T^{2} \) |
| 67 | \( 1 - 152. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 28.0T + 3.57e5T^{2} \) |
| 73 | \( 1 - 124.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 748. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 348.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 416. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.50e3T + 9.12e5T^{2} \) |
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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
−11.68530284848850518155846659456, −10.33996913371115381515524808249, −9.361894351889647957170356248508, −8.484025124408220648750141306349, −7.69439201612487070610764708186, −6.38139627187857872278603891384, −5.89993652261902059051029438803, −4.35678596922442963672341785204, −3.62958155783184590405366833243, −1.26938629402932056024796356638,
0.75598928085712460918537129922, 2.22531172187213723127903786493, 3.55386899246269021964876044584, 4.50575952736159761610702950658, 5.83546413146800133873819062549, 6.88541912577441794963254659822, 8.454821697395534294414855233963, 9.145762725638194270689087955939, 10.06216774239612074429162179747, 11.10114737649566903096253564103