L(s) = 1 | + 303. i·3-s − 3.49e4i·5-s + (−4.15e5 − 7.11e5i)7-s + 4.69e6·9-s + 1.67e7·11-s − 1.13e8i·13-s + 1.06e7·15-s − 7.03e8i·17-s − 5.27e8i·19-s + (2.15e8 − 1.26e8i)21-s − 3.75e9·23-s − 1.22e9·25-s + 2.87e9i·27-s + 7.44e9·29-s − 4.18e10i·31-s + ⋯ |
L(s) = 1 | + 0.138i·3-s − 0.447i·5-s + (−0.504 − 0.863i)7-s + 0.980·9-s + 0.859·11-s − 1.81i·13-s + 0.0620·15-s − 1.71i·17-s − 0.589i·19-s + (0.119 − 0.0700i)21-s − 1.10·23-s − 0.199·25-s + 0.274i·27-s + 0.431·29-s − 1.52i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.863 + 0.504i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.863 + 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{15}{2})\) |
\(\approx\) |
\(2.268730213\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.268730213\) |
\(L(8)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 3.49e4iT \) |
| 7 | \( 1 + (4.15e5 + 7.11e5i)T \) |
good | 3 | \( 1 - 303. iT - 4.78e6T^{2} \) |
| 11 | \( 1 - 1.67e7T + 3.79e14T^{2} \) |
| 13 | \( 1 + 1.13e8iT - 3.93e15T^{2} \) |
| 17 | \( 1 + 7.03e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 + 5.27e8iT - 7.99e17T^{2} \) |
| 23 | \( 1 + 3.75e9T + 1.15e19T^{2} \) |
| 29 | \( 1 - 7.44e9T + 2.97e20T^{2} \) |
| 31 | \( 1 + 4.18e10iT - 7.56e20T^{2} \) |
| 37 | \( 1 - 1.57e11T + 9.01e21T^{2} \) |
| 41 | \( 1 - 1.56e10iT - 3.79e22T^{2} \) |
| 43 | \( 1 - 3.48e11T + 7.38e22T^{2} \) |
| 47 | \( 1 + 5.96e11iT - 2.56e23T^{2} \) |
| 53 | \( 1 + 1.35e12T + 1.37e24T^{2} \) |
| 59 | \( 1 - 2.06e11iT - 6.19e24T^{2} \) |
| 61 | \( 1 - 5.42e12iT - 9.87e24T^{2} \) |
| 67 | \( 1 + 3.85e12T + 3.67e25T^{2} \) |
| 71 | \( 1 - 1.27e13T + 8.27e25T^{2} \) |
| 73 | \( 1 - 5.67e12iT - 1.22e26T^{2} \) |
| 79 | \( 1 - 2.24e13T + 3.68e26T^{2} \) |
| 83 | \( 1 + 8.87e12iT - 7.36e26T^{2} \) |
| 89 | \( 1 - 5.93e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 + 2.97e13iT - 6.52e27T^{2} \) |
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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
−9.931482708596419196254715480448, −9.450886274514263424371791543599, −7.921730166943333993454334465104, −7.16381389564760076691434126102, −5.95995411930423049081554655833, −4.67531122912577060145586364006, −3.83529936486656104617764627018, −2.61036578504982359379994871346, −0.938799567666015940450308011539, −0.50974187987669089069062025602,
1.41947490237912237601728851879, 2.09615659084227893315169531964, 3.63205609170766884661537029046, 4.42092790963092128443584398769, 6.25034001562252605465232528650, 6.49968565946847836630586882403, 7.909684283986000999127117895021, 9.106988792925720179614152049009, 9.837067402725183896668633507978, 10.99853895849208295651979141626