| L(s) = 1 | + 48·3-s + 625·5-s + 532·7-s − 1.73e4·9-s + 3.31e4·11-s − 9.96e4·13-s + 3.00e4·15-s − 4.43e5·17-s + 3.57e5·19-s + 2.55e4·21-s + 1.42e5·23-s + 3.90e5·25-s − 1.77e6·27-s + 1.52e6·29-s − 7.32e6·31-s + 1.59e6·33-s + 3.32e5·35-s − 2.66e6·37-s − 4.78e6·39-s − 7.93e6·41-s + 2.11e7·43-s − 1.08e7·45-s − 1.60e7·47-s − 4.00e7·49-s − 2.12e7·51-s − 8.78e7·53-s + 2.07e7·55-s + ⋯ |
| L(s) = 1 | + 0.342·3-s + 0.447·5-s + 0.0837·7-s − 0.882·9-s + 0.683·11-s − 0.967·13-s + 0.153·15-s − 1.28·17-s + 0.628·19-s + 0.0286·21-s + 0.106·23-s + 1/5·25-s − 0.644·27-s + 0.401·29-s − 1.42·31-s + 0.233·33-s + 0.0374·35-s − 0.233·37-s − 0.331·39-s − 0.438·41-s + 0.944·43-s − 0.394·45-s − 0.480·47-s − 0.992·49-s − 0.440·51-s − 1.52·53-s + 0.305·55-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 | \( 1 - 16 p T + p^{9} T^{2} \) |
| 7 | \( 1 - 76 p T + p^{9} T^{2} \) |
| 11 | \( 1 - 33180 T + p^{9} T^{2} \) |
| 13 | \( 1 + 99682 T + p^{9} T^{2} \) |
| 17 | \( 1 + 443454 T + p^{9} T^{2} \) |
| 19 | \( 1 - 357244 T + p^{9} T^{2} \) |
| 23 | \( 1 - 142956 T + p^{9} T^{2} \) |
| 29 | \( 1 - 1527966 T + p^{9} T^{2} \) |
| 31 | \( 1 + 7323416 T + p^{9} T^{2} \) |
| 37 | \( 1 + 2666842 T + p^{9} T^{2} \) |
| 41 | \( 1 + 7939014 T + p^{9} T^{2} \) |
| 43 | \( 1 - 21174520 T + p^{9} T^{2} \) |
| 47 | \( 1 + 16059636 T + p^{9} T^{2} \) |
| 53 | \( 1 + 87822234 T + p^{9} T^{2} \) |
| 59 | \( 1 + 120625212 T + p^{9} T^{2} \) |
| 61 | \( 1 - 93576542 T + p^{9} T^{2} \) |
| 67 | \( 1 + 193621688 T + p^{9} T^{2} \) |
| 71 | \( 1 + 417763488 T + p^{9} T^{2} \) |
| 73 | \( 1 + 450372742 T + p^{9} T^{2} \) |
| 79 | \( 1 - 91425472 T + p^{9} T^{2} \) |
| 83 | \( 1 - 652637376 T + p^{9} T^{2} \) |
| 89 | \( 1 + 170059206 T + p^{9} T^{2} \) |
| 97 | \( 1 + 10947022 T + p^{9} T^{2} \) |
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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
−11.94513896201162300454794078721, −10.93716964058327753834731091042, −9.531937786019079619010650388315, −8.766539823592261452667866416202, −7.35940706425456786645423700267, −6.07478317018659445088387524096, −4.73395138676371559004778660895, −3.09616033905009160801014147013, −1.83624011284592308795873799211, 0,
1.83624011284592308795873799211, 3.09616033905009160801014147013, 4.73395138676371559004778660895, 6.07478317018659445088387524096, 7.35940706425456786645423700267, 8.766539823592261452667866416202, 9.531937786019079619010650388315, 10.93716964058327753834731091042, 11.94513896201162300454794078721
