| L(s) = 1 | − 15.5i·3-s − 195.·5-s − 277. i·7-s − 243·9-s − 1.75e3i·11-s + 1.24e3·13-s + 3.04e3i·15-s + 6.88e3·17-s − 4.40e3i·19-s − 4.32e3·21-s − 1.27e4i·23-s + 2.24e4·25-s + 3.78e3i·27-s + 8.27e3·29-s + 4.39e4i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.56·5-s − 0.809i·7-s − 0.333·9-s − 1.31i·11-s + 0.567·13-s + 0.901i·15-s + 1.40·17-s − 0.641i·19-s − 0.467·21-s − 1.04i·23-s + 1.43·25-s + 0.192i·27-s + 0.339·29-s + 1.47i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.819828768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.819828768\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 15.5iT \) |
| good | 5 | \( 1 + 195.T + 1.56e4T^{2} \) |
| 7 | \( 1 + 277. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 1.75e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 1.24e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 6.88e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 4.40e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 1.27e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 8.27e3T + 5.94e8T^{2} \) |
| 31 | \( 1 - 4.39e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 1.21e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 5.47e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 4.54e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 1.52e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 2.72e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + 2.13e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 8.39e4T + 5.15e10T^{2} \) |
| 67 | \( 1 - 3.73e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 6.67e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 3.99e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 4.35e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 2.46e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 8.09e4T + 4.96e11T^{2} \) |
| 97 | \( 1 - 8.77e5T + 8.32e11T^{2} \) |
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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
−8.772929256559580500864329735898, −8.254718980389314666599359470408, −7.51575473873837529341980184469, −6.79320616622726973110330280727, −5.75323569402012494654559632792, −4.50436889574082154094132627340, −3.59683230051603668636990128057, −2.91868918129261995771020215067, −0.923699138764194389445154183346, −0.69325744961555115068155294958,
0.70745467213097553604770313850, 2.18232909772393217786445751928, 3.50309172322668316420766911647, 4.01113462458049741280102485654, 5.07545388978270240271406570681, 5.94368986535116143654058528776, 7.34244862393461499263358564620, 7.83543840708870416421289268949, 8.723064228327657597002656462748, 9.625497478264584788208573993759
