L(s) = 1 | + 27i·3-s − 69.8i·5-s + 860.·7-s − 729·9-s + 6.10e3i·11-s − 5.33e3i·13-s + 1.88e3·15-s + 3.86e4·17-s − 1.35e4i·19-s + 2.32e4i·21-s − 9.96e4·23-s + 7.32e4·25-s − 1.96e4i·27-s − 8.11e4i·29-s + 1.53e5·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 0.250i·5-s + 0.947·7-s − 0.333·9-s + 1.38i·11-s − 0.673i·13-s + 0.144·15-s + 1.90·17-s − 0.454i·19-s + 0.547i·21-s − 1.70·23-s + 0.937·25-s − 0.192i·27-s − 0.618i·29-s + 0.922·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(2.696736284\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.696736284\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 27iT \) |
good | 5 | \( 1 + 69.8iT - 7.81e4T^{2} \) |
| 7 | \( 1 - 860.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 6.10e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 + 5.33e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 3.86e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.35e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 9.96e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 8.11e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 1.53e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.67e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 3.84e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.50e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 5.03e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.61e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 1.58e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 2.61e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 1.52e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 1.15e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.52e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 7.64e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.34e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 1.70e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.70e6T + 8.07e13T^{2} \) |
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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
−9.952371391439779339866248391988, −9.761489935590850949030931508977, −8.124912646319523804036413480408, −7.893209923073739030105107327298, −6.40296012243668294851028809894, −5.13846652632572681810687183172, −4.64218575670161950032730126519, −3.37885786435595855973513588789, −2.05269452421564668918993536401, −0.872899795729505294207254685799,
0.73458187609075718150336185087, 1.63867205317007382070271855141, 2.92366085247958399269290819395, 4.04154485525338483877717246423, 5.47472098873521621220246489279, 6.12118202339675846420240115800, 7.41945935581569372382415461920, 8.087969027707307793089826720752, 8.898386358805547763563205642280, 10.19571086819454050468808247693