| L(s) = 1 | + 16·2-s − 41·3-s + 256·4-s − 1.03e3·5-s − 656·6-s − 3.48e3·7-s + 4.09e3·8-s − 1.80e4·9-s − 1.66e4·10-s + 1.46e4·11-s − 1.04e4·12-s − 1.99e5·13-s − 5.57e4·14-s + 4.25e4·15-s + 6.55e4·16-s + 1.64e5·17-s − 2.88e5·18-s − 2.77e5·19-s − 2.65e5·20-s + 1.42e5·21-s + 2.34e5·22-s − 1.21e6·23-s − 1.67e5·24-s − 8.73e5·25-s − 3.19e6·26-s + 1.54e6·27-s − 8.91e5·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.292·3-s + 1/2·4-s − 0.743·5-s − 0.206·6-s − 0.548·7-s + 0.353·8-s − 0.914·9-s − 0.525·10-s + 0.301·11-s − 0.146·12-s − 1.94·13-s − 0.387·14-s + 0.217·15-s + 1/4·16-s + 0.476·17-s − 0.646·18-s − 0.488·19-s − 0.371·20-s + 0.160·21-s + 0.213·22-s − 0.902·23-s − 0.103·24-s − 0.447·25-s − 1.37·26-s + 0.559·27-s − 0.274·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{4} T \) |
| 11 | \( 1 - p^{4} T \) |
| good | 3 | \( 1 + 41 T + p^{9} T^{2} \) |
| 5 | \( 1 + 1039 T + p^{9} T^{2} \) |
| 7 | \( 1 + 3482 T + p^{9} T^{2} \) |
| 13 | \( 1 + 199796 T + p^{9} T^{2} \) |
| 17 | \( 1 - 164038 T + p^{9} T^{2} \) |
| 19 | \( 1 + 277560 T + p^{9} T^{2} \) |
| 23 | \( 1 + 1211721 T + p^{9} T^{2} \) |
| 29 | \( 1 - 4248880 T + p^{9} T^{2} \) |
| 31 | \( 1 - 9112927 T + p^{9} T^{2} \) |
| 37 | \( 1 - 10500403 T + p^{9} T^{2} \) |
| 41 | \( 1 + 844768 T + p^{9} T^{2} \) |
| 43 | \( 1 - 586 p^{2} T + p^{9} T^{2} \) |
| 47 | \( 1 + 45843752 T + p^{9} T^{2} \) |
| 53 | \( 1 - 5568394 T + p^{9} T^{2} \) |
| 59 | \( 1 + 106773315 T + p^{9} T^{2} \) |
| 61 | \( 1 + 98810468 T + p^{9} T^{2} \) |
| 67 | \( 1 + 168277647 T + p^{9} T^{2} \) |
| 71 | \( 1 - 67984277 T + p^{9} T^{2} \) |
| 73 | \( 1 + 65392116 T + p^{9} T^{2} \) |
| 79 | \( 1 - 85785910 T + p^{9} T^{2} \) |
| 83 | \( 1 + 103589846 T + p^{9} T^{2} \) |
| 89 | \( 1 + 809499425 T + p^{9} T^{2} \) |
| 97 | \( 1 - 859612633 T + p^{9} T^{2} \) |
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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
−15.11648964559451572110160594055, −14.06814743663093316411542823472, −12.35114962314464925151293435271, −11.69615424213164573458859382922, −9.974315244635573273233001577104, −7.915773484469194114366567105042, −6.31214862959737422331184478945, −4.62718890687948201702639967109, −2.84891623328249360793969739685, 0,
2.84891623328249360793969739685, 4.62718890687948201702639967109, 6.31214862959737422331184478945, 7.915773484469194114366567105042, 9.974315244635573273233001577104, 11.69615424213164573458859382922, 12.35114962314464925151293435271, 14.06814743663093316411542823472, 15.11648964559451572110160594055
