| L(s) = 1 | + 8i·2-s − 27i·3-s − 64·4-s + 216·6-s − 391i·7-s − 512i·8-s − 729·9-s − 4.39e3·11-s + 1.72e3i·12-s + 1.34e4i·13-s + 3.12e3·14-s + 4.09e3·16-s − 7.68e3i·17-s − 5.83e3i·18-s + 1.37e4·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.430i·7-s − 0.353i·8-s − 0.333·9-s − 0.996·11-s + 0.288i·12-s + 1.69i·13-s + 0.304·14-s + 0.250·16-s − 0.379i·17-s − 0.235i·18-s + 0.458·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.618834736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.618834736\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8iT \) |
| 3 | \( 1 + 27iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 391iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 4.39e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.34e4iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 7.68e3iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 1.37e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.54e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 - 1.57e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 9.93e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.61e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 5.21e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.40e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 5.08e4iT - 5.06e11T^{2} \) |
| 53 | \( 1 - 8.91e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 1.34e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 3.39e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.24e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 2.73e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.02e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 1.57e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.79e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 5.80e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.49e6iT - 8.07e13T^{2} \) |
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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
−11.90677760786218491724298961118, −10.75026838569408361694920302226, −9.504950163020247138574651077647, −8.453387167078567504802141829034, −7.36339014793102259309513929543, −6.65218933767494223036784120148, −5.34760769385700164135427999730, −4.12644510167032123690826295023, −2.37418218062608443963522803983, −0.73982100135571957176577749057,
0.69533343277918214669614747826, 2.50810715866138551598039632187, 3.45130182480188653390703064844, 4.97291923013368513672036060904, 5.77619748568146310290896091864, 7.73475764715182246712930746004, 8.624979881973633056590674474981, 9.896978957823366808642658464928, 10.49337007390083128974833537127, 11.48519671570809561493621341679
