| L(s) = 1 | − 815. i·3-s − 270·5-s − 1.81e5i·7-s − 1.33e5·9-s + 1.65e6i·11-s − 4.38e6·13-s + 2.20e5i·15-s − 3.93e7·17-s − 5.42e7i·19-s − 1.47e8·21-s + 2.40e8i·23-s − 2.44e8·25-s − 3.24e8i·27-s + 1.63e8·29-s − 2.31e8i·31-s + ⋯ |
| L(s) = 1 | − 1.11i·3-s − 0.0172·5-s − 1.53i·7-s − 0.251·9-s + 0.932i·11-s − 0.908·13-s + 0.0193i·15-s − 1.63·17-s − 1.15i·19-s − 1.72·21-s + 1.62i·23-s − 0.999·25-s − 0.837i·27-s + 0.274·29-s − 0.260i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(-1.02025i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(-1.02025i\) |
| \(L(7)\) |
|
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 | \( 1 + 815. iT - 5.31e5T^{2} \) |
| 5 | \( 1 + 270T + 2.44e8T^{2} \) |
| 7 | \( 1 + 1.81e5iT - 1.38e10T^{2} \) |
| 11 | \( 1 - 1.65e6iT - 3.13e12T^{2} \) |
| 13 | \( 1 + 4.38e6T + 2.32e13T^{2} \) |
| 17 | \( 1 + 3.93e7T + 5.82e14T^{2} \) |
| 19 | \( 1 + 5.42e7iT - 2.21e15T^{2} \) |
| 23 | \( 1 - 2.40e8iT - 2.19e16T^{2} \) |
| 29 | \( 1 - 1.63e8T + 3.53e17T^{2} \) |
| 31 | \( 1 + 2.31e8iT - 7.87e17T^{2} \) |
| 37 | \( 1 - 3.60e9T + 6.58e18T^{2} \) |
| 41 | \( 1 - 2.12e9T + 2.25e19T^{2} \) |
| 43 | \( 1 + 2.60e9iT - 3.99e19T^{2} \) |
| 47 | \( 1 + 1.75e10iT - 1.16e20T^{2} \) |
| 53 | \( 1 + 1.35e10T + 4.91e20T^{2} \) |
| 59 | \( 1 + 2.48e10iT - 1.77e21T^{2} \) |
| 61 | \( 1 - 3.54e10T + 2.65e21T^{2} \) |
| 67 | \( 1 + 1.21e11iT - 8.18e21T^{2} \) |
| 71 | \( 1 - 1.24e11iT - 1.64e22T^{2} \) |
| 73 | \( 1 + 5.98e9T + 2.29e22T^{2} \) |
| 79 | \( 1 + 1.21e11iT - 5.90e22T^{2} \) |
| 83 | \( 1 + 3.97e11iT - 1.06e23T^{2} \) |
| 89 | \( 1 + 7.53e11T + 2.46e23T^{2} \) |
| 97 | \( 1 + 9.70e11T + 6.93e23T^{2} \) |
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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
−15.42285661434096535589549387815, −13.71403536040581059576772715477, −13.00412984598497790545783145346, −11.42032499966870580239066793270, −9.771190881328347893332020980107, −7.55394315475801950121053792152, −6.87293260096239717638981542024, −4.39878940698416530283590930981, −1.98842836974668574194191951257, −0.39631622499006013507911395808,
2.56132212862914461330099792769, 4.46653623063737179605744498951, 5.98811764880138173666071788211, 8.506759102513945019756856288995, 9.655142482550109840001759991771, 11.14361209533333589836052312272, 12.57005163851365104825117544890, 14.53424765393267501546454662761, 15.54901918209113364478999916766, 16.45173641579911494963432989282
