L(s) = 1 | + 5.07·3-s + 5.51i·5-s − 8.74·7-s + 16.7·9-s + 0.103i·11-s + 7.16·13-s + 28.0i·15-s + 24.4·17-s + (−7.40 + 17.4i)19-s − 44.3·21-s + 10.9·23-s − 5.44·25-s + 39.3·27-s − 12.9·29-s + 40.6i·31-s + ⋯ |
L(s) = 1 | + 1.69·3-s + 1.10i·5-s − 1.24·7-s + 1.86·9-s + 0.00938i·11-s + 0.551·13-s + 1.86i·15-s + 1.43·17-s + (−0.389 + 0.920i)19-s − 2.11·21-s + 0.473·23-s − 0.217·25-s + 1.45·27-s − 0.447·29-s + 1.31i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.138 - 0.990i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(3.157023221\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.157023221\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (7.40 - 17.4i)T \) |
good | 3 | \( 1 - 5.07T + 9T^{2} \) |
| 5 | \( 1 - 5.51iT - 25T^{2} \) |
| 7 | \( 1 + 8.74T + 49T^{2} \) |
| 11 | \( 1 - 0.103iT - 121T^{2} \) |
| 13 | \( 1 - 7.16T + 169T^{2} \) |
| 17 | \( 1 - 24.4T + 289T^{2} \) |
| 23 | \( 1 - 10.9T + 529T^{2} \) |
| 29 | \( 1 + 12.9T + 841T^{2} \) |
| 31 | \( 1 - 40.6iT - 961T^{2} \) |
| 37 | \( 1 + 42.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + 1.13iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 63.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 17.7T + 2.20e3T^{2} \) |
| 53 | \( 1 - 52.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + 117.T + 3.48e3T^{2} \) |
| 61 | \( 1 - 26.0iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 67.7T + 4.48e3T^{2} \) |
| 71 | \( 1 - 92.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 20.5T + 5.32e3T^{2} \) |
| 79 | \( 1 - 72.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 69.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 8.48iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 130. iT - 9.40e3T^{2} \) |
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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
−9.796058892299236045956288869903, −8.883998163864456141307537466508, −8.161835354419526582290387202636, −7.28041840157814841449683484307, −6.70615162028500870040403954989, −5.70302582202992710709081959159, −4.01636479483646955987537796489, −3.14242410662371443927404726226, −3.01659152152379004977804049202, −1.56186268852009801365139706786,
0.75283686235746203201260727124, 2.08537343739554704109520592304, 3.23201161906721490179109226728, 3.76414281638230858639229648596, 4.91266299162025492583425283576, 6.04437164168209386032807464652, 7.15377080175984882695750598366, 7.87866471022414037249018100518, 8.815897493343640651754843126281, 9.102791300246178362034537178465