L(s) = 1 | + 1.55e5i·3-s + 8.02e6i·5-s − 7.41e8·7-s − 1.36e10·9-s − 5.97e10i·11-s + 4.24e11i·13-s − 1.24e12·15-s + 1.25e13·17-s + 1.26e13i·19-s − 1.15e14i·21-s + 2.08e14·23-s + 4.12e14·25-s − 4.97e14i·27-s + 5.67e14i·29-s + 5.21e15·31-s + ⋯ |
L(s) = 1 | + 1.51i·3-s + 0.367i·5-s − 0.992·7-s − 1.30·9-s − 0.694i·11-s + 0.853i·13-s − 0.558·15-s + 1.51·17-s + 0.472i·19-s − 1.50i·21-s + 1.04·23-s + 0.864·25-s − 0.464i·27-s + 0.250i·29-s + 1.14·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(11)\) |
\(\approx\) |
\(2.072786179\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.072786179\) |
\(L(\frac{23}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 - 1.55e5iT - 1.04e10T^{2} \) |
| 5 | \( 1 - 8.02e6iT - 4.76e14T^{2} \) |
| 7 | \( 1 + 7.41e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 5.97e10iT - 7.40e21T^{2} \) |
| 13 | \( 1 - 4.24e11iT - 2.47e23T^{2} \) |
| 17 | \( 1 - 1.25e13T + 6.90e25T^{2} \) |
| 19 | \( 1 - 1.26e13iT - 7.14e26T^{2} \) |
| 23 | \( 1 - 2.08e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 5.67e14iT - 5.13e30T^{2} \) |
| 31 | \( 1 - 5.21e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 4.07e16iT - 8.55e32T^{2} \) |
| 41 | \( 1 + 2.32e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 1.35e15iT - 2.00e34T^{2} \) |
| 47 | \( 1 - 3.31e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 7.09e17iT - 1.62e36T^{2} \) |
| 59 | \( 1 + 3.66e18iT - 1.54e37T^{2} \) |
| 61 | \( 1 - 7.38e18iT - 3.10e37T^{2} \) |
| 67 | \( 1 - 4.59e18iT - 2.22e38T^{2} \) |
| 71 | \( 1 - 4.10e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 4.51e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 1.57e20T + 7.08e39T^{2} \) |
| 83 | \( 1 - 1.74e20iT - 1.99e40T^{2} \) |
| 89 | \( 1 + 2.32e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 8.12e20T + 5.27e41T^{2} \) |
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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
−11.11388237496692646170868030363, −10.17935927248731785033776325468, −9.530427365058920030282179957716, −8.486142715646929868031167121014, −6.85326248654056921486845999253, −5.74138897537967004169155554384, −4.62467888397012139853877785247, −3.44674452900058253032301899378, −2.96577214506439002262893582280, −0.981671208773898540598205002572,
0.52450555803526151594954005580, 1.04770564609002644217879576162, 2.37667402346182673440783160137, 3.32030924021423305553372078043, 5.10156327302366386321872728267, 6.23036449940639569391764896540, 7.15003947356937709493605435349, 7.960102716144685389342412685787, 9.224062545775011537037721499583, 10.39952734568705383447980230366