L(s) = 1 | + 3.49e3·5-s − 5.54e4·7-s + 5.97e5·11-s + 1.37e6·13-s − 1.01e7·17-s − 7.29e6·19-s + 3.20e7·23-s − 3.66e7·25-s + 1.36e7·29-s + 2.33e8·31-s − 1.93e8·35-s − 2.57e8·37-s + 2.21e8·41-s − 1.69e9·43-s − 5.27e8·47-s + 1.09e9·49-s − 3.27e9·53-s + 2.08e9·55-s + 3.00e9·59-s − 1.16e10·61-s + 4.79e9·65-s − 1.71e10·67-s − 2.61e10·71-s − 7.03e9·73-s − 3.31e10·77-s − 4.19e9·79-s + 3.97e10·83-s + ⋯ |
L(s) = 1 | + 0.499·5-s − 1.24·7-s + 1.11·11-s + 1.02·13-s − 1.73·17-s − 0.676·19-s + 1.03·23-s − 0.750·25-s + 0.123·29-s + 1.46·31-s − 0.622·35-s − 0.611·37-s + 0.298·41-s − 1.76·43-s − 0.335·47-s + 0.555·49-s − 1.07·53-s + 0.558·55-s + 0.546·59-s − 1.76·61-s + 0.512·65-s − 1.55·67-s − 1.72·71-s − 0.397·73-s − 1.39·77-s − 0.153·79-s + 1.10·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 698 p T + p^{11} T^{2} \) |
| 7 | \( 1 + 55464 T + p^{11} T^{2} \) |
| 11 | \( 1 - 597004 T + p^{11} T^{2} \) |
| 13 | \( 1 - 1373878 T + p^{11} T^{2} \) |
| 17 | \( 1 + 10140850 T + p^{11} T^{2} \) |
| 19 | \( 1 + 7297396 T + p^{11} T^{2} \) |
| 23 | \( 1 - 32057464 T + p^{11} T^{2} \) |
| 29 | \( 1 - 13605402 T + p^{11} T^{2} \) |
| 31 | \( 1 - 233160800 T + p^{11} T^{2} \) |
| 37 | \( 1 + 6967194 p T + p^{11} T^{2} \) |
| 41 | \( 1 - 221438598 T + p^{11} T^{2} \) |
| 43 | \( 1 + 1697758892 T + p^{11} T^{2} \) |
| 47 | \( 1 + 527509392 T + p^{11} T^{2} \) |
| 53 | \( 1 + 3277379822 T + p^{11} T^{2} \) |
| 59 | \( 1 - 3001908988 T + p^{11} T^{2} \) |
| 61 | \( 1 + 11630023610 T + p^{11} T^{2} \) |
| 67 | \( 1 + 17189000548 T + p^{11} T^{2} \) |
| 71 | \( 1 + 26169539608 T + p^{11} T^{2} \) |
| 73 | \( 1 + 7039021094 T + p^{11} T^{2} \) |
| 79 | \( 1 + 4199910416 T + p^{11} T^{2} \) |
| 83 | \( 1 - 39739936436 T + p^{11} T^{2} \) |
| 89 | \( 1 + 10565331594 T + p^{11} T^{2} \) |
| 97 | \( 1 + 69851645662 T + p^{11} T^{2} \) |
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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
−11.79862511465648457274650231111, −10.62128131277090014794656483418, −9.430583013471741892744439912246, −8.655754776372209917409313847386, −6.66853640703438522848840159542, −6.23126239224997448463342937829, −4.36486952022787405286851382575, −3.08826482265313388913654896999, −1.56271838474469756894641894194, 0,
1.56271838474469756894641894194, 3.08826482265313388913654896999, 4.36486952022787405286851382575, 6.23126239224997448463342937829, 6.66853640703438522848840159542, 8.655754776372209917409313847386, 9.430583013471741892744439912246, 10.62128131277090014794656483418, 11.79862511465648457274650231111