L(s) = 1 | − 14i·3-s + (−55 + 10i)5-s − 158i·7-s + 47·9-s − 148·11-s + 684i·13-s + (140 + 770i)15-s + 2.04e3i·17-s − 2.22e3·19-s − 2.21e3·21-s − 1.24e3i·23-s + (2.92e3 − 1.10e3i)25-s − 4.06e3i·27-s − 270·29-s + 2.04e3·31-s + ⋯ |
L(s) = 1 | − 0.898i·3-s + (−0.983 + 0.178i)5-s − 1.21i·7-s + 0.193·9-s − 0.368·11-s + 1.12i·13-s + (0.160 + 0.883i)15-s + 1.71i·17-s − 1.41·19-s − 1.09·21-s − 0.491i·23-s + (0.936 − 0.352i)25-s − 1.07i·27-s − 0.0596·29-s + 0.382·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.197210845\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.197210845\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (55 - 10i)T \) |
good | 3 | \( 1 + 14iT - 243T^{2} \) |
| 7 | \( 1 + 158iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 148T + 1.61e5T^{2} \) |
| 13 | \( 1 - 684iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 2.04e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.22e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.24e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 270T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.04e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.37e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 2.39e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.29e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.06e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.96e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 3.97e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.22e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.20e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 4.24e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.01e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 3.52e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.78e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 8.52e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.72e4iT - 8.58e9T^{2} \) |
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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
−10.80512971703842205938793926893, −10.20389991219384000273201360489, −8.554766010370562719683013998795, −7.895379098747612100151775831001, −6.93452499489299452121806688560, −6.42339261506598745320188060853, −4.41709671029085249603609144419, −3.86324907561574315849105982650, −2.07406856841339630670662715508, −0.860149954218277697006783331640,
0.42311220523008900598544120931, 2.51131129197342711917014645541, 3.60940092228080351029101369613, 4.79269969644412968960353032309, 5.47686317546994207955442921791, 7.01254228920753954371332083765, 8.112614617917449204419180571065, 8.905186250823924820764183138716, 9.821021895269934300427676916367, 10.78438816917617883692052374153