| L(s) = 1 | + 3·3-s + 6·5-s + 16·7-s + 9·9-s − 12·11-s + 38·13-s + 18·15-s − 126·17-s − 20·19-s + 48·21-s − 168·23-s − 89·25-s + 27·27-s + 30·29-s + 88·31-s − 36·33-s + 96·35-s + 254·37-s + 114·39-s + 42·41-s + 52·43-s + 54·45-s + 96·47-s − 87·49-s − 378·51-s + 198·53-s − 72·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.536·5-s + 0.863·7-s + 1/3·9-s − 0.328·11-s + 0.810·13-s + 0.309·15-s − 1.79·17-s − 0.241·19-s + 0.498·21-s − 1.52·23-s − 0.711·25-s + 0.192·27-s + 0.192·29-s + 0.509·31-s − 0.189·33-s + 0.463·35-s + 1.12·37-s + 0.468·39-s + 0.159·41-s + 0.184·43-s + 0.178·45-s + 0.297·47-s − 0.253·49-s − 1.03·51-s + 0.513·53-s − 0.176·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.740630162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.740630162\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| good | 5 | \( 1 - 6 T + p^{3} T^{2} \) |
| 7 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 38 T + p^{3} T^{2} \) |
| 17 | \( 1 + 126 T + p^{3} T^{2} \) |
| 19 | \( 1 + 20 T + p^{3} T^{2} \) |
| 23 | \( 1 + 168 T + p^{3} T^{2} \) |
| 29 | \( 1 - 30 T + p^{3} T^{2} \) |
| 31 | \( 1 - 88 T + p^{3} T^{2} \) |
| 37 | \( 1 - 254 T + p^{3} T^{2} \) |
| 41 | \( 1 - 42 T + p^{3} T^{2} \) |
| 43 | \( 1 - 52 T + p^{3} T^{2} \) |
| 47 | \( 1 - 96 T + p^{3} T^{2} \) |
| 53 | \( 1 - 198 T + p^{3} T^{2} \) |
| 59 | \( 1 - 660 T + p^{3} T^{2} \) |
| 61 | \( 1 + 538 T + p^{3} T^{2} \) |
| 67 | \( 1 + 884 T + p^{3} T^{2} \) |
| 71 | \( 1 + 792 T + p^{3} T^{2} \) |
| 73 | \( 1 - 218 T + p^{3} T^{2} \) |
| 79 | \( 1 - 520 T + p^{3} T^{2} \) |
| 83 | \( 1 - 492 T + p^{3} T^{2} \) |
| 89 | \( 1 - 810 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1154 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.10691688359171568921321810275, −13.91635818657304001672664390565, −13.21198275139265964661150152458, −11.57402572492926697932355447577, −10.37838089010935976399949088931, −8.964906868441660068959189039192, −7.908886243795708851278077883517, −6.16857799865124706908386024437, −4.34483398041061295087593373294, −2.10167815592214400200405597643,
2.10167815592214400200405597643, 4.34483398041061295087593373294, 6.16857799865124706908386024437, 7.908886243795708851278077883517, 8.964906868441660068959189039192, 10.37838089010935976399949088931, 11.57402572492926697932355447577, 13.21198275139265964661150152458, 13.91635818657304001672664390565, 15.10691688359171568921321810275
