| L(s) = 1 | + 32·2-s + 236·3-s + 1.02e3·4-s − 3.12e3·5-s + 7.55e3·6-s + 3.33e4·7-s + 3.27e4·8-s − 3.35e3·9-s − 1.00e5·10-s + 2.41e5·12-s + 1.06e6·14-s − 7.37e5·15-s + 1.04e6·16-s − 1.07e5·18-s − 3.20e6·20-s + 7.87e6·21-s − 1.16e6·23-s + 7.73e6·24-s + 9.76e6·25-s − 1.47e7·27-s + 3.41e7·28-s − 3.81e7·29-s − 2.36e7·30-s + 3.35e7·32-s − 1.04e8·35-s − 3.43e6·36-s − 1.02e8·40-s + ⋯ |
| L(s) = 1 | + 2-s + 0.971·3-s + 4-s − 5-s + 0.971·6-s + 1.98·7-s + 8-s − 0.0567·9-s − 10-s + 0.971·12-s + 1.98·14-s − 0.971·15-s + 16-s − 0.0567·18-s − 20-s + 1.92·21-s − 0.181·23-s + 0.971·24-s + 25-s − 1.02·27-s + 1.98·28-s − 1.86·29-s − 0.971·30-s + 32-s − 1.98·35-s − 0.0567·36-s − 40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(4.405783231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.405783231\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{5} T \) |
| 5 | \( 1 + p^{5} T \) |
| good | 3 | \( 1 - 236 T + p^{10} T^{2} \) |
| 7 | \( 1 - 33364 T + p^{10} T^{2} \) |
| 11 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 13 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 17 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 19 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 23 | \( 1 + 1169564 T + p^{10} T^{2} \) |
| 29 | \( 1 + 38179702 T + p^{10} T^{2} \) |
| 31 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 37 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 41 | \( 1 + 211028098 T + p^{10} T^{2} \) |
| 43 | \( 1 + 223663364 T + p^{10} T^{2} \) |
| 47 | \( 1 - 96887764 T + p^{10} T^{2} \) |
| 53 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 59 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 61 | \( 1 + 1041591898 T + p^{10} T^{2} \) |
| 67 | \( 1 - 2343243964 T + p^{10} T^{2} \) |
| 71 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 73 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 79 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 83 | \( 1 - 5449159036 T + p^{10} T^{2} \) |
| 89 | \( 1 - 11118190898 T + p^{10} T^{2} \) |
| 97 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
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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
−15.17793374074356893957452987076, −14.74459576918435170788564610535, −13.63841284483698823339377217860, −11.88611632429736112097981534716, −11.04276095483687753955919945165, −8.395260008323433536398239261502, −7.53092184813561796448379420770, −5.03564739332827385465573965688, −3.65492227822618844801339424105, −1.89315891206400814180533219189,
1.89315891206400814180533219189, 3.65492227822618844801339424105, 5.03564739332827385465573965688, 7.53092184813561796448379420770, 8.395260008323433536398239261502, 11.04276095483687753955919945165, 11.88611632429736112097981534716, 13.63841284483698823339377217860, 14.74459576918435170788564610535, 15.17793374074356893957452987076
