| L(s) = 1 | + 16·2-s + 81·3-s + 256·4-s + 1.29e3·6-s − 4.27e3·7-s + 4.09e3·8-s + 6.56e3·9-s − 1.53e4·11-s + 2.07e4·12-s − 1.74e4·13-s − 6.84e4·14-s + 6.55e4·16-s − 5.17e5·17-s + 1.04e5·18-s + 4.04e5·19-s − 3.46e5·21-s − 2.45e5·22-s − 3.31e5·23-s + 3.31e5·24-s − 2.79e5·26-s + 5.31e5·27-s − 1.09e6·28-s − 4.31e6·29-s − 1.25e6·31-s + 1.04e6·32-s − 1.24e6·33-s − 8.27e6·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.673·7-s + 0.353·8-s + 1/3·9-s − 0.315·11-s + 0.288·12-s − 0.169·13-s − 0.476·14-s + 1/4·16-s − 1.50·17-s + 0.235·18-s + 0.712·19-s − 0.388·21-s − 0.223·22-s − 0.246·23-s + 0.204·24-s − 0.119·26-s + 0.192·27-s − 0.336·28-s − 1.13·29-s − 0.244·31-s + 0.176·32-s − 0.182·33-s − 1.06·34-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 | \( 1 + 611 p T + p^{9} T^{2} \) |
| 11 | \( 1 + 15318 T + p^{9} T^{2} \) |
| 13 | \( 1 + 17441 T + p^{9} T^{2} \) |
| 17 | \( 1 + 30426 p T + p^{9} T^{2} \) |
| 19 | \( 1 - 404675 T + p^{9} T^{2} \) |
| 23 | \( 1 + 331086 T + p^{9} T^{2} \) |
| 29 | \( 1 + 4313550 T + p^{9} T^{2} \) |
| 31 | \( 1 + 1256443 T + p^{9} T^{2} \) |
| 37 | \( 1 + 10224722 T + p^{9} T^{2} \) |
| 41 | \( 1 + 17362128 T + p^{9} T^{2} \) |
| 43 | \( 1 + 8621321 T + p^{9} T^{2} \) |
| 47 | \( 1 - 28968798 T + p^{9} T^{2} \) |
| 53 | \( 1 + 16225236 T + p^{9} T^{2} \) |
| 59 | \( 1 - 6025110 T + p^{9} T^{2} \) |
| 61 | \( 1 + 9966793 T + p^{9} T^{2} \) |
| 67 | \( 1 - 173779243 T + p^{9} T^{2} \) |
| 71 | \( 1 + 152168928 T + p^{9} T^{2} \) |
| 73 | \( 1 + 347721986 T + p^{9} T^{2} \) |
| 79 | \( 1 - 30159200 T + p^{9} T^{2} \) |
| 83 | \( 1 + 308273766 T + p^{9} T^{2} \) |
| 89 | \( 1 - 958503840 T + p^{9} T^{2} \) |
| 97 | \( 1 - 38982223 T + p^{9} T^{2} \) |
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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
−10.87148822609170086542872296927, −9.772590319230467826635820857485, −8.750278015930386624376664835602, −7.45226565242957670943389757359, −6.53192871102983797820706326791, −5.24345251540056511144096311426, −3.99423951062058295059265863023, −2.96140030333909449484701033422, −1.84181691525244183450870831161, 0,
1.84181691525244183450870831161, 2.96140030333909449484701033422, 3.99423951062058295059265863023, 5.24345251540056511144096311426, 6.53192871102983797820706326791, 7.45226565242957670943389757359, 8.750278015930386624376664835602, 9.772590319230467826635820857485, 10.87148822609170086542872296927
