| L(s) = 1 | − 4.83e3·5-s + 1.67e4·7-s + 5.34e5·11-s − 5.77e5·13-s + 6.90e6·17-s − 1.06e7·19-s + 1.86e7·23-s − 2.54e7·25-s − 1.28e8·29-s + 5.28e7·31-s − 8.08e7·35-s − 1.82e8·37-s − 3.08e8·41-s + 1.71e7·43-s + 2.68e9·47-s − 1.69e9·49-s + 1.59e9·53-s − 2.58e9·55-s − 5.18e9·59-s + 6.95e9·61-s + 2.79e9·65-s + 1.54e10·67-s + 9.79e9·71-s + 1.46e9·73-s + 8.95e9·77-s − 3.81e10·79-s − 2.93e10·83-s + ⋯ |
| L(s) = 1 | − 0.691·5-s + 0.376·7-s + 1.00·11-s − 0.431·13-s + 1.17·17-s − 0.987·19-s + 0.603·23-s − 0.522·25-s − 1.16·29-s + 0.331·31-s − 0.260·35-s − 0.431·37-s − 0.415·41-s + 0.0177·43-s + 1.70·47-s − 0.858·49-s + 0.524·53-s − 0.691·55-s − 0.944·59-s + 1.05·61-s + 0.298·65-s + 1.40·67-s + 0.644·71-s + 0.0826·73-s + 0.376·77-s − 1.39·79-s − 0.817·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.905551392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.905551392\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + 966 p T + p^{11} T^{2} \) |
| 7 | \( 1 - 2392 p T + p^{11} T^{2} \) |
| 11 | \( 1 - 534612 T + p^{11} T^{2} \) |
| 13 | \( 1 + 577738 T + p^{11} T^{2} \) |
| 17 | \( 1 - 6905934 T + p^{11} T^{2} \) |
| 19 | \( 1 + 10661420 T + p^{11} T^{2} \) |
| 23 | \( 1 - 18643272 T + p^{11} T^{2} \) |
| 29 | \( 1 + 128406630 T + p^{11} T^{2} \) |
| 31 | \( 1 - 52843168 T + p^{11} T^{2} \) |
| 37 | \( 1 + 182213314 T + p^{11} T^{2} \) |
| 41 | \( 1 + 308120442 T + p^{11} T^{2} \) |
| 43 | \( 1 - 17125708 T + p^{11} T^{2} \) |
| 47 | \( 1 - 2687348496 T + p^{11} T^{2} \) |
| 53 | \( 1 - 1596055698 T + p^{11} T^{2} \) |
| 59 | \( 1 + 5189203740 T + p^{11} T^{2} \) |
| 61 | \( 1 - 6956478662 T + p^{11} T^{2} \) |
| 67 | \( 1 - 15481826884 T + p^{11} T^{2} \) |
| 71 | \( 1 - 9791485272 T + p^{11} T^{2} \) |
| 73 | \( 1 - 1463791322 T + p^{11} T^{2} \) |
| 79 | \( 1 + 38116845680 T + p^{11} T^{2} \) |
| 83 | \( 1 + 29335099668 T + p^{11} T^{2} \) |
| 89 | \( 1 - 24992917110 T + p^{11} T^{2} \) |
| 97 | \( 1 - 75013568546 T + p^{11} T^{2} \) |
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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
−11.17568950211061757190005506264, −9.993032662241137468116381524359, −8.890835397672664011799457990637, −7.87468758812402576353911492337, −6.93914345311736080979609866504, −5.63600385338659860741306060606, −4.36274710812613638648672824029, −3.43855916468361630446313473643, −1.90557158865858241740105272759, −0.66074785596446848386226949725,
0.66074785596446848386226949725, 1.90557158865858241740105272759, 3.43855916468361630446313473643, 4.36274710812613638648672824029, 5.63600385338659860741306060606, 6.93914345311736080979609866504, 7.87468758812402576353911492337, 8.890835397672664011799457990637, 9.993032662241137468116381524359, 11.17568950211061757190005506264
