| L(s) = 1 | − 738·3-s − 3.12e3·5-s − 2.55e4·7-s + 3.67e5·9-s − 7.69e5·11-s − 9.18e5·13-s + 2.30e6·15-s + 1.03e7·17-s + 5.52e6·19-s + 1.88e7·21-s + 3.99e7·23-s + 9.76e6·25-s − 1.40e8·27-s − 1.52e7·29-s + 2.41e8·31-s + 5.67e8·33-s + 7.99e7·35-s − 2.57e7·37-s + 6.78e8·39-s − 1.21e9·41-s + 6.83e8·43-s − 1.14e9·45-s − 1.53e9·47-s − 1.32e9·49-s − 7.61e9·51-s + 3.57e9·53-s + 2.40e9·55-s + ⋯ |
| L(s) = 1 | − 1.75·3-s − 0.447·5-s − 0.575·7-s + 2.07·9-s − 1.43·11-s − 0.686·13-s + 0.784·15-s + 1.76·17-s + 0.511·19-s + 1.00·21-s + 1.29·23-s + 1/5·25-s − 1.88·27-s − 0.138·29-s + 1.51·31-s + 2.52·33-s + 0.257·35-s − 0.0610·37-s + 1.20·39-s − 1.64·41-s + 0.708·43-s − 0.927·45-s − 0.977·47-s − 0.669·49-s − 3.08·51-s + 1.17·53-s + 0.643·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + p^{5} T \) |
| good | 3 | \( 1 + 82 p^{2} T + p^{11} T^{2} \) |
| 7 | \( 1 + 25574 T + p^{11} T^{2} \) |
| 11 | \( 1 + 769152 T + p^{11} T^{2} \) |
| 13 | \( 1 + 918982 T + p^{11} T^{2} \) |
| 17 | \( 1 - 10312794 T + p^{11} T^{2} \) |
| 19 | \( 1 - 5521660 T + p^{11} T^{2} \) |
| 23 | \( 1 - 39973422 T + p^{11} T^{2} \) |
| 29 | \( 1 + 15269010 T + p^{11} T^{2} \) |
| 31 | \( 1 - 241583788 T + p^{11} T^{2} \) |
| 37 | \( 1 + 25751446 T + p^{11} T^{2} \) |
| 41 | \( 1 + 1217700138 T + p^{11} T^{2} \) |
| 43 | \( 1 - 683436262 T + p^{11} T^{2} \) |
| 47 | \( 1 + 1537395294 T + p^{11} T^{2} \) |
| 53 | \( 1 - 3572891298 T + p^{11} T^{2} \) |
| 59 | \( 1 - 1069039020 T + p^{11} T^{2} \) |
| 61 | \( 1 + 2091535078 T + p^{11} T^{2} \) |
| 67 | \( 1 - 1462369186 T + p^{11} T^{2} \) |
| 71 | \( 1 + 9660178332 T + p^{11} T^{2} \) |
| 73 | \( 1 + 5603447662 T + p^{11} T^{2} \) |
| 79 | \( 1 + 5026936280 T + p^{11} T^{2} \) |
| 83 | \( 1 - 38405955462 T + p^{11} T^{2} \) |
| 89 | \( 1 - 35558583210 T + p^{11} T^{2} \) |
| 97 | \( 1 - 10572232514 T + p^{11} T^{2} \) |
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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
−11.69870006225678329414667392409, −10.51239987478212167572527655066, −9.876194534853667217910086918410, −7.83015326331146330134466641523, −6.83607939626386919999102483922, −5.53250665588736119705594328191, −4.86617041692149178913517812453, −3.09255581517835965678352253160, −0.968419550133429213382222494728, 0,
0.968419550133429213382222494728, 3.09255581517835965678352253160, 4.86617041692149178913517812453, 5.53250665588736119705594328191, 6.83607939626386919999102483922, 7.83015326331146330134466641523, 9.876194534853667217910086918410, 10.51239987478212167572527655066, 11.69870006225678329414667392409
