| L(s) = 1 | − 170·3-s + 544·5-s + 2.40e3·7-s + 9.21e3·9-s − 4.88e4·11-s − 1.58e4·13-s − 9.24e4·15-s − 2.14e4·17-s + 7.16e5·19-s − 4.08e5·21-s + 2.47e6·23-s − 1.65e6·25-s + 1.77e6·27-s + 5.55e6·29-s − 5.79e6·31-s + 8.30e6·33-s + 1.30e6·35-s − 3.89e6·37-s + 2.69e6·39-s − 6.36e6·41-s + 1.87e7·43-s + 5.01e6·45-s − 5.65e7·47-s + 5.76e6·49-s + 3.64e6·51-s − 5.98e7·53-s − 2.65e7·55-s + ⋯ |
| L(s) = 1 | − 1.21·3-s + 0.389·5-s + 0.377·7-s + 0.468·9-s − 1.00·11-s − 0.154·13-s − 0.471·15-s − 0.0621·17-s + 1.26·19-s − 0.457·21-s + 1.84·23-s − 0.848·25-s + 0.644·27-s + 1.45·29-s − 1.12·31-s + 1.21·33-s + 0.147·35-s − 0.341·37-s + 0.186·39-s − 0.351·41-s + 0.834·43-s + 0.182·45-s − 1.69·47-s + 1/7·49-s + 0.0753·51-s − 1.04·53-s − 0.391·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - p^{4} T \) |
| good | 3 | \( 1 + 170 T + p^{9} T^{2} \) |
| 5 | \( 1 - 544 T + p^{9} T^{2} \) |
| 11 | \( 1 + 48824 T + p^{9} T^{2} \) |
| 13 | \( 1 + 15876 T + p^{9} T^{2} \) |
| 17 | \( 1 + 21418 T + p^{9} T^{2} \) |
| 19 | \( 1 - 716410 T + p^{9} T^{2} \) |
| 23 | \( 1 - 2470000 T + p^{9} T^{2} \) |
| 29 | \( 1 - 5556826 T + p^{9} T^{2} \) |
| 31 | \( 1 + 5799348 T + p^{9} T^{2} \) |
| 37 | \( 1 + 3894430 T + p^{9} T^{2} \) |
| 41 | \( 1 + 6360858 T + p^{9} T^{2} \) |
| 43 | \( 1 - 18701296 T + p^{9} T^{2} \) |
| 47 | \( 1 + 56539068 T + p^{9} T^{2} \) |
| 53 | \( 1 + 59894682 T + p^{9} T^{2} \) |
| 59 | \( 1 + 165629662 T + p^{9} T^{2} \) |
| 61 | \( 1 - 51419016 T + p^{9} T^{2} \) |
| 67 | \( 1 + 93546508 T + p^{9} T^{2} \) |
| 71 | \( 1 - 95633536 T + p^{9} T^{2} \) |
| 73 | \( 1 - 306496402 T + p^{9} T^{2} \) |
| 79 | \( 1 + 496474152 T + p^{9} T^{2} \) |
| 83 | \( 1 - 371486962 T + p^{9} T^{2} \) |
| 89 | \( 1 + 165482550 T + p^{9} T^{2} \) |
| 97 | \( 1 - 758016742 T + p^{9} T^{2} \) |
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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
−11.24634019882665133174372321152, −10.53571305011329054391452373867, −9.367200443341473620836649155807, −7.901172528417452573373137647567, −6.70773018414231265974451156717, −5.46792867779799244919365520459, −4.90119905664388820552365278285, −2.93313359030907023241727494025, −1.27665855404393310956080943253, 0,
1.27665855404393310956080943253, 2.93313359030907023241727494025, 4.90119905664388820552365278285, 5.46792867779799244919365520459, 6.70773018414231265974451156717, 7.901172528417452573373137647567, 9.367200443341473620836649155807, 10.53571305011329054391452373867, 11.24634019882665133174372321152
