| L(s) = 1 | + 4·2-s − 81·3-s − 496·4-s − 324·6-s + 7.68e3·7-s − 4.03e3·8-s + 6.56e3·9-s − 8.64e4·11-s + 4.01e4·12-s + 1.49e5·13-s + 3.07e4·14-s + 2.37e5·16-s + 2.07e5·17-s + 2.62e4·18-s + 7.16e5·19-s − 6.22e5·21-s − 3.45e5·22-s − 1.36e6·23-s + 3.26e5·24-s + 5.99e5·26-s − 5.31e5·27-s − 3.80e6·28-s − 3.19e6·29-s − 2.34e6·31-s + 3.01e6·32-s + 6.99e6·33-s + 8.30e5·34-s + ⋯ |
| L(s) = 1 | + 0.176·2-s − 0.577·3-s − 0.968·4-s − 0.102·6-s + 1.20·7-s − 0.348·8-s + 1/3·9-s − 1.77·11-s + 0.559·12-s + 1.45·13-s + 0.213·14-s + 0.907·16-s + 0.602·17-s + 0.0589·18-s + 1.26·19-s − 0.698·21-s − 0.314·22-s − 1.02·23-s + 0.200·24-s + 0.257·26-s − 0.192·27-s − 1.17·28-s − 0.838·29-s − 0.456·31-s + 0.508·32-s + 1.02·33-s + 0.106·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + p^{4} T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - p^{2} T + p^{9} T^{2} \) |
| 7 | \( 1 - 7680 T + p^{9} T^{2} \) |
| 11 | \( 1 + 86404 T + p^{9} T^{2} \) |
| 13 | \( 1 - 149978 T + p^{9} T^{2} \) |
| 17 | \( 1 - 207622 T + p^{9} T^{2} \) |
| 19 | \( 1 - 716284 T + p^{9} T^{2} \) |
| 23 | \( 1 + 1369920 T + p^{9} T^{2} \) |
| 29 | \( 1 + 3194402 T + p^{9} T^{2} \) |
| 31 | \( 1 + 2349000 T + p^{9} T^{2} \) |
| 37 | \( 1 + 18735710 T + p^{9} T^{2} \) |
| 41 | \( 1 + 29282630 T + p^{9} T^{2} \) |
| 43 | \( 1 - 1516724 T + p^{9} T^{2} \) |
| 47 | \( 1 + 615752 T + p^{9} T^{2} \) |
| 53 | \( 1 + 4747430 T + p^{9} T^{2} \) |
| 59 | \( 1 - 60616076 T + p^{9} T^{2} \) |
| 61 | \( 1 + 126745682 T + p^{9} T^{2} \) |
| 67 | \( 1 - 111182652 T + p^{9} T^{2} \) |
| 71 | \( 1 + 175551608 T + p^{9} T^{2} \) |
| 73 | \( 1 - 61233350 T + p^{9} T^{2} \) |
| 79 | \( 1 - 234431160 T + p^{9} T^{2} \) |
| 83 | \( 1 + 118910388 T + p^{9} T^{2} \) |
| 89 | \( 1 + 316534326 T + p^{9} T^{2} \) |
| 97 | \( 1 + 242912258 T + p^{9} T^{2} \) |
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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
−12.14000545484851769286267346000, −10.99940786811770618800709694424, −10.04547076247559752499367549344, −8.503366002916459423130559048385, −7.68904773216773760603003673193, −5.61209039109175492722977477842, −5.05769520030952163570621721897, −3.57503024227007667797924384252, −1.45182853782726453189915728514, 0,
1.45182853782726453189915728514, 3.57503024227007667797924384252, 5.05769520030952163570621721897, 5.61209039109175492722977477842, 7.68904773216773760603003673193, 8.503366002916459423130559048385, 10.04547076247559752499367549344, 10.99940786811770618800709694424, 12.14000545484851769286267346000
