| L(s) = 1 | − 16·2-s − 81·3-s + 256·4-s + 1.29e3·6-s − 2.40e3·7-s − 4.09e3·8-s + 6.56e3·9-s + 1.48e4·11-s − 2.07e4·12-s − 3.02e4·13-s + 3.85e4·14-s + 6.55e4·16-s − 1.01e5·17-s − 1.04e5·18-s + 1.60e5·19-s + 1.95e5·21-s − 2.38e5·22-s − 5.26e5·23-s + 3.31e5·24-s + 4.84e5·26-s − 5.31e5·27-s − 6.16e5·28-s + 1.78e6·29-s − 7.06e5·31-s − 1.04e6·32-s − 1.20e6·33-s + 1.62e6·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.379·7-s − 0.353·8-s + 1/3·9-s + 0.306·11-s − 0.288·12-s − 0.293·13-s + 0.268·14-s + 1/4·16-s − 0.295·17-s − 0.235·18-s + 0.283·19-s + 0.218·21-s − 0.216·22-s − 0.392·23-s + 0.204·24-s + 0.207·26-s − 0.192·27-s − 0.189·28-s + 0.469·29-s − 0.137·31-s − 0.176·32-s − 0.177·33-s + 0.208·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{4} T \) |
| 3 | \( 1 + p^{4} T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 344 p T + p^{9} T^{2} \) |
| 11 | \( 1 - 14892 T + p^{9} T^{2} \) |
| 13 | \( 1 + 30254 T + p^{9} T^{2} \) |
| 17 | \( 1 + 101778 T + p^{9} T^{2} \) |
| 19 | \( 1 - 160820 T + p^{9} T^{2} \) |
| 23 | \( 1 + 526584 T + p^{9} T^{2} \) |
| 29 | \( 1 - 1788030 T + p^{9} T^{2} \) |
| 31 | \( 1 + 706528 T + p^{9} T^{2} \) |
| 37 | \( 1 - 8889082 T + p^{9} T^{2} \) |
| 41 | \( 1 + 10313238 T + p^{9} T^{2} \) |
| 43 | \( 1 - 27839956 T + p^{9} T^{2} \) |
| 47 | \( 1 - 54742512 T + p^{9} T^{2} \) |
| 53 | \( 1 - 101510826 T + p^{9} T^{2} \) |
| 59 | \( 1 + 118394340 T + p^{9} T^{2} \) |
| 61 | \( 1 - 178661342 T + p^{9} T^{2} \) |
| 67 | \( 1 + 244239428 T + p^{9} T^{2} \) |
| 71 | \( 1 - 81740232 T + p^{9} T^{2} \) |
| 73 | \( 1 + 277364234 T + p^{9} T^{2} \) |
| 79 | \( 1 + 140711920 T + p^{9} T^{2} \) |
| 83 | \( 1 - 422051436 T + p^{9} T^{2} \) |
| 89 | \( 1 + 753422310 T + p^{9} T^{2} \) |
| 97 | \( 1 + 1041114338 T + p^{9} T^{2} \) |
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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
−10.69402666658881339192979577311, −9.811346076485456883720543363445, −8.863824564939339332957951471444, −7.59242249382040585956507875359, −6.62517029431898230621805529323, −5.59709568704745765815612282522, −4.12734421395238664844569267177, −2.58161947944092157073725522323, −1.14853923089011548666789126501, 0,
1.14853923089011548666789126501, 2.58161947944092157073725522323, 4.12734421395238664844569267177, 5.59709568704745765815612282522, 6.62517029431898230621805529323, 7.59242249382040585956507875359, 8.863824564939339332957951471444, 9.811346076485456883720543363445, 10.69402666658881339192979577311
