| L(s) = 1 | − 9.74e4·11-s − 8.23e5·17-s − 3.35e6·19-s + 9.76e6·25-s + 3.77e7·41-s − 2.14e8·43-s + 2.82e8·49-s + 9.21e8·59-s − 1.81e9·67-s − 1.60e9·73-s + 9.60e7·83-s + 1.11e10·89-s − 9.87e9·97-s + 2.74e10·107-s − 2.88e10·113-s + ⋯ |
| L(s) = 1 | − 0.604·11-s − 0.580·17-s − 1.35·19-s + 25-s + 0.326·41-s − 1.45·43-s + 49-s + 1.28·59-s − 1.34·67-s − 0.774·73-s + 0.0243·83-s + 1.99·89-s − 1.14·97-s + 1.96·107-s − 1.56·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(1.539752908\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.539752908\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 7 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 11 | \( 1 + 97426 T + p^{10} T^{2} \) |
| 13 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 17 | \( 1 + 823682 T + p^{10} T^{2} \) |
| 19 | \( 1 + 3353726 T + p^{10} T^{2} \) |
| 23 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 29 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 31 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 37 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 41 | \( 1 - 37778926 T + p^{10} T^{2} \) |
| 43 | \( 1 + 214485614 T + p^{10} T^{2} \) |
| 47 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 53 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 59 | \( 1 - 921043598 T + p^{10} T^{2} \) |
| 61 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 67 | \( 1 + 1813708382 T + p^{10} T^{2} \) |
| 71 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 73 | \( 1 + 1605781582 T + p^{10} T^{2} \) |
| 79 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 83 | \( 1 - 96051518 T + p^{10} T^{2} \) |
| 89 | \( 1 - 11116019374 T + p^{10} T^{2} \) |
| 97 | \( 1 + 9872978014 T + p^{10} T^{2} \) |
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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
−10.23174917817097704132387200810, −8.997642362541336893120637068858, −8.284923892782079449622615556105, −7.14465774716899679712754706727, −6.24676004658103149080287814027, −5.09517477600305482506258284604, −4.14799805436231887604603874332, −2.87185188095309228767915767922, −1.88297659535103519488378485087, −0.51343917301733733145802683475,
0.51343917301733733145802683475, 1.88297659535103519488378485087, 2.87185188095309228767915767922, 4.14799805436231887604603874332, 5.09517477600305482506258284604, 6.24676004658103149080287814027, 7.14465774716899679712754706727, 8.284923892782079449622615556105, 8.997642362541336893120637068858, 10.23174917817097704132387200810
