| L(s) = 1 | + 17·2-s − 6.52e4·4-s + 3.90e5·5-s − 9.88e6·7-s − 2.22e6·8-s + 4.30e7·9-s + 6.64e6·10-s + 2.14e8·11-s − 6.78e8·13-s − 1.68e8·14-s + 4.23e9·16-s − 1.39e10·17-s + 7.31e8·18-s − 2.54e10·20-s + 3.64e9·22-s + 1.52e11·25-s − 1.15e10·26-s + 6.45e11·28-s − 1.20e12·31-s + 2.17e11·32-s − 2.36e11·34-s − 3.86e12·35-s − 2.80e12·36-s − 8.68e11·40-s + 7.61e12·43-s − 1.39e13·44-s + 1.68e13·45-s + ⋯ |
| L(s) = 1 | + 0.0664·2-s − 0.995·4-s + 5-s − 1.71·7-s − 0.132·8-s + 9-s + 0.0664·10-s + 11-s − 0.831·13-s − 0.113·14-s + 0.986·16-s − 1.99·17-s + 0.0664·18-s − 0.995·20-s + 0.0664·22-s + 25-s − 0.0551·26-s + 1.70·28-s − 1.41·31-s + 0.198·32-s − 0.132·34-s − 1.71·35-s − 0.995·36-s − 0.132·40-s + 0.651·43-s − 0.995·44-s + 45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{17}{2})\) |
\(\approx\) |
\(1.365566759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.365566759\) |
| \(L(9)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - p^{8} T \) |
| 11 | \( 1 - p^{8} T \) |
| good | 2 | \( 1 - 17 T + p^{16} T^{2} \) |
| 3 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 7 | \( 1 + 9886078 T + p^{16} T^{2} \) |
| 13 | \( 1 + 678010558 T + p^{16} T^{2} \) |
| 17 | \( 1 + 13921943038 T + p^{16} T^{2} \) |
| 19 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 23 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 29 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 31 | \( 1 + 1206552215038 T + p^{16} T^{2} \) |
| 37 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 41 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 43 | \( 1 - 7613774646722 T + p^{16} T^{2} \) |
| 47 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 53 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 59 | \( 1 + 140214236988478 T + p^{16} T^{2} \) |
| 61 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 67 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 71 | \( 1 + 77726196639358 T + p^{16} T^{2} \) |
| 73 | \( 1 - 1564720076407682 T + p^{16} T^{2} \) |
| 79 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 83 | \( 1 - 3613022253130562 T + p^{16} T^{2} \) |
| 89 | \( 1 - 7841390882244482 T + p^{16} T^{2} \) |
| 97 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
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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
−12.51830525686830427120796360906, −10.47573271264795019051967085761, −9.331010952439187844539511244087, −9.252296985072002914751056796739, −6.99529871006252066130842109265, −6.16803222757185707985165383814, −4.69085571308426156154189960273, −3.59908520764849132093685835239, −2.08611279208284574806216158746, −0.56601966709087005834708799516,
0.56601966709087005834708799516, 2.08611279208284574806216158746, 3.59908520764849132093685835239, 4.69085571308426156154189960273, 6.16803222757185707985165383814, 6.99529871006252066130842109265, 9.252296985072002914751056796739, 9.331010952439187844539511244087, 10.47573271264795019051967085761, 12.51830525686830427120796360906
