| L(s) = 1 | + 204·3-s + 625·5-s − 5.43e3·7-s + 2.19e4·9-s − 7.39e4·11-s − 1.14e5·13-s + 1.27e5·15-s + 4.16e4·17-s − 1.05e6·19-s − 1.10e6·21-s − 1.59e6·23-s + 3.90e5·25-s + 4.59e5·27-s + 2.18e6·29-s + 9.61e6·31-s − 1.50e7·33-s − 3.39e6·35-s + 4.79e6·37-s − 2.33e7·39-s + 9.53e6·41-s + 1.34e7·43-s + 1.37e7·45-s − 1.14e7·47-s − 1.08e7·49-s + 8.50e6·51-s + 5.36e7·53-s − 4.62e7·55-s + ⋯ |
| L(s) = 1 | + 1.45·3-s + 0.447·5-s − 0.855·7-s + 1.11·9-s − 1.52·11-s − 1.11·13-s + 0.650·15-s + 0.121·17-s − 1.86·19-s − 1.24·21-s − 1.19·23-s + 1/5·25-s + 0.166·27-s + 0.573·29-s + 1.87·31-s − 2.21·33-s − 0.382·35-s + 0.421·37-s − 1.61·39-s + 0.526·41-s + 0.600·43-s + 0.498·45-s − 0.342·47-s − 0.268·49-s + 0.176·51-s + 0.933·53-s − 0.680·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - p^{4} T \) |
| good | 3 | \( 1 - 68 p T + p^{9} T^{2} \) |
| 7 | \( 1 + 776 p T + p^{9} T^{2} \) |
| 11 | \( 1 + 73932 T + p^{9} T^{2} \) |
| 13 | \( 1 + 114514 T + p^{9} T^{2} \) |
| 17 | \( 1 - 41682 T + p^{9} T^{2} \) |
| 19 | \( 1 + 1057460 T + p^{9} T^{2} \) |
| 23 | \( 1 + 1599336 T + p^{9} T^{2} \) |
| 29 | \( 1 - 2184510 T + p^{9} T^{2} \) |
| 31 | \( 1 - 9619648 T + p^{9} T^{2} \) |
| 37 | \( 1 - 4799942 T + p^{9} T^{2} \) |
| 41 | \( 1 - 9531882 T + p^{9} T^{2} \) |
| 43 | \( 1 - 13464484 T + p^{9} T^{2} \) |
| 47 | \( 1 + 11441952 T + p^{9} T^{2} \) |
| 53 | \( 1 - 53615766 T + p^{9} T^{2} \) |
| 59 | \( 1 + 81862620 T + p^{9} T^{2} \) |
| 61 | \( 1 + 104691298 T + p^{9} T^{2} \) |
| 67 | \( 1 + 2098076 p T + p^{9} T^{2} \) |
| 71 | \( 1 + 97098792 T + p^{9} T^{2} \) |
| 73 | \( 1 - 171848906 T + p^{9} T^{2} \) |
| 79 | \( 1 - 117380080 T + p^{9} T^{2} \) |
| 83 | \( 1 + 323637636 T + p^{9} T^{2} \) |
| 89 | \( 1 + 894379110 T + p^{9} T^{2} \) |
| 97 | \( 1 - 232678562 T + p^{9} T^{2} \) |
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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
−12.42934237763799951212056984893, −10.37131710355605936260390695013, −9.763972546196520475806975697300, −8.528222667281485992275981306611, −7.65521220020599115408414540802, −6.19492378195620288775281631842, −4.46740088568152008898193492567, −2.85207143831355368773919282654, −2.26613149917669568652689170921, 0,
2.26613149917669568652689170921, 2.85207143831355368773919282654, 4.46740088568152008898193492567, 6.19492378195620288775281631842, 7.65521220020599115408414540802, 8.528222667281485992275981306611, 9.763972546196520475806975697300, 10.37131710355605936260390695013, 12.42934237763799951212056984893
