| L(s) = 1 | − 5.70i·2-s − 8.30·3-s + 95.3·4-s + 279. i·5-s + 47.4i·6-s − 343i·7-s − 1.27e3i·8-s − 2.11e3·9-s + 1.59e3·10-s − 1.42e3i·11-s − 792.·12-s + (−7.91e3 − 177. i)13-s − 1.95e3·14-s − 2.32e3i·15-s + 4.92e3·16-s + 3.38e4·17-s + ⋯ |
| L(s) = 1 | − 0.504i·2-s − 0.177·3-s + 0.745·4-s + 1.00i·5-s + 0.0896i·6-s − 0.377i·7-s − 0.880i·8-s − 0.968·9-s + 0.504·10-s − 0.323i·11-s − 0.132·12-s + (−0.999 − 0.0224i)13-s − 0.190·14-s − 0.177i·15-s + 0.300·16-s + 1.66·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0224 + 0.999i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.0224 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.30015 - 1.32967i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30015 - 1.32967i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 343iT \) |
| 13 | \( 1 + (7.91e3 + 177. i)T \) |
| good | 2 | \( 1 + 5.70iT - 128T^{2} \) |
| 3 | \( 1 + 8.30T + 2.18e3T^{2} \) |
| 5 | \( 1 - 279. iT - 7.81e4T^{2} \) |
| 11 | \( 1 + 1.42e3iT - 1.94e7T^{2} \) |
| 17 | \( 1 - 3.38e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.73e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 - 9.83e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.68e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.26e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 - 3.45e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 6.67e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 - 2.95e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.34e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 - 2.42e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 8.86e4iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 1.03e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.37e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 7.26e5iT - 9.09e12T^{2} \) |
| 73 | \( 1 - 5.05e5iT - 1.10e13T^{2} \) |
| 79 | \( 1 - 2.01e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.73e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 8.61e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 + 2.98e5iT - 8.07e13T^{2} \) |
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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
−12.07770931452306500845973829700, −11.21037684520062442851190122102, −10.60382634790439193062008109901, −9.392853880437185865305762809117, −7.56951446552639411447703897889, −6.78444916325667082087562788907, −5.41498508872509282939844871157, −3.31675694115307781214360343334, −2.54081504757771573330589317663, −0.61779304142186072201014336189,
1.36480101812499720051739592169, 2.98807189623200908784942363641, 5.13775633220311279204856264742, 5.78715819827048683794177173044, 7.35366604658014477995053340931, 8.312878270668923838347944179166, 9.510905866055202361201420835924, 10.91511102608381811660862606170, 12.13437778628336758491866920374, 12.55168728237057482687961385866
