L(s) = 1 | + (−0.664 + 2.74i)2-s + (−7.11 − 3.65i)4-s + (14.2 + 8.25i)5-s + (19.2 − 11.1i)7-s + (14.7 − 17.1i)8-s + (−32.1 + 33.8i)10-s + (6.37 + 11.0i)11-s + (−11.1 + 19.3i)13-s + (17.7 + 60.3i)14-s + (37.3 + 51.9i)16-s + 117. i·17-s − 27.7i·19-s + (−71.5 − 110. i)20-s + (−34.6 + 10.1i)22-s + (17.5 − 30.4i)23-s + ⋯ |
L(s) = 1 | + (−0.234 + 0.972i)2-s + (−0.889 − 0.456i)4-s + (1.27 + 0.737i)5-s + (1.04 − 0.600i)7-s + (0.652 − 0.757i)8-s + (−1.01 + 1.06i)10-s + (0.174 + 0.302i)11-s + (−0.238 + 0.413i)13-s + (0.339 + 1.15i)14-s + (0.583 + 0.812i)16-s + 1.67i·17-s − 0.334i·19-s + (−0.800 − 1.24i)20-s + (−0.335 + 0.0988i)22-s + (0.159 − 0.276i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0267 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0267 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.22648 + 1.19405i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.22648 + 1.19405i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.664 - 2.74i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-14.2 - 8.25i)T + (62.5 + 108. i)T^{2} \) |
| 7 | \( 1 + (-19.2 + 11.1i)T + (171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-6.37 - 11.0i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (11.1 - 19.3i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 - 117. iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 27.7iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-17.5 + 30.4i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (1.01 - 0.584i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (119. + 68.9i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 233.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (13.2 + 7.65i)T + (3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-361. + 208. i)T + (3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (116. + 201. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + 180. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (313. - 543. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (382. + 661. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (113. + 65.5i)T + (1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 22.6T + 3.57e5T^{2} \) |
| 73 | \( 1 - 387.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (486. - 280. i)T + (2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (342. + 592. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 278. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (264. + 458. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−13.87386124105517342963900899960, −12.84843315766551462259341212396, −10.96021313602547441827481959016, −10.19805858180164658186292733947, −9.112403413814073445840614955101, −7.82445459380083098410569824267, −6.72084448942522084415863973255, −5.71854327891820513412418388038, −4.31302868995231844262584449880, −1.73697533590655057560116691915,
1.25741966306243671265690169112, 2.59684262565933651842884293654, 4.77285444104095791947096409121, 5.60939029169106377891142846389, 7.82481001933996095899107423110, 9.037537310888077187317629253646, 9.595842563155207537499299640056, 10.92234871975839329456265504867, 11.88247927324929178777994959231, 12.86163934812073691112268959971