L(s) = 1 | + (240. − 345. i)3-s − 113.·5-s + (4.40e4 + 6.36e3i)7-s + (−6.15e4 − 1.66e5i)9-s + 4.71e5i·11-s + 1.52e6i·13-s + (−2.72e4 + 3.91e4i)15-s + 1.63e6·17-s + 2.97e6i·19-s + (1.27e7 − 1.36e7i)21-s + 2.79e7i·23-s − 4.88e7·25-s + (−7.21e7 − 1.86e7i)27-s + 1.72e8i·29-s + 6.77e7i·31-s + ⋯ |
L(s) = 1 | + (0.571 − 0.820i)3-s − 0.0162·5-s + (0.989 + 0.143i)7-s + (−0.347 − 0.937i)9-s + 0.883i·11-s + 1.13i·13-s + (−0.00926 + 0.0133i)15-s + 0.278·17-s + 0.275i·19-s + (0.682 − 0.730i)21-s + 0.904i·23-s − 0.999·25-s + (−0.968 − 0.250i)27-s + 1.56i·29-s + 0.424i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.682 - 0.730i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.682 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(2.21358 + 0.960836i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.21358 + 0.960836i\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-240. + 345. i)T \) |
| 7 | \( 1 + (-4.40e4 - 6.36e3i)T \) |
good | 5 | \( 1 + 113.T + 4.88e7T^{2} \) |
| 11 | \( 1 - 4.71e5iT - 2.85e11T^{2} \) |
| 13 | \( 1 - 1.52e6iT - 1.79e12T^{2} \) |
| 17 | \( 1 - 1.63e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 2.97e6iT - 1.16e14T^{2} \) |
| 23 | \( 1 - 2.79e7iT - 9.52e14T^{2} \) |
| 29 | \( 1 - 1.72e8iT - 1.22e16T^{2} \) |
| 31 | \( 1 - 6.77e7iT - 2.54e16T^{2} \) |
| 37 | \( 1 + 2.11e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.13e9T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.22e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 2.73e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 4.10e9iT - 9.26e18T^{2} \) |
| 59 | \( 1 - 9.33e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 9.27e9iT - 4.35e19T^{2} \) |
| 67 | \( 1 - 9.31e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.13e9iT - 2.31e20T^{2} \) |
| 73 | \( 1 - 1.18e10iT - 3.13e20T^{2} \) |
| 79 | \( 1 - 2.57e9T + 7.47e20T^{2} \) |
| 83 | \( 1 - 3.83e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 4.88e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.23e11iT - 7.15e21T^{2} \) |
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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
−12.12850392061473735160086846043, −11.40953639372210924331438647246, −9.773470714250000555229112053533, −8.689150750021338825042672945853, −7.65576460938972289225525905328, −6.73992321257894834197213270918, −5.17760992782570718892487749473, −3.73549669198767920091346668864, −2.08341608268798208000877026336, −1.38604522358929284425634662337,
0.54033576663538540849204812645, 2.25781998269837513103315019185, 3.52334715976568590971790553402, 4.72430531351554344494227952224, 5.84447290887853002706976508173, 7.84489414250270849760044854594, 8.385321801811815825978831121469, 9.747011829897564416442128155164, 10.74098738225083540167761143375, 11.59423589469467509627199080796