| L(s) = 1 | − 4·11-s − 54·13-s + 114·17-s + 44·19-s + 96·23-s − 134·29-s − 272·31-s + 98·37-s + 6·41-s − 12·43-s − 200·47-s − 343·49-s + 654·53-s − 36·59-s − 442·61-s + 188·67-s + 632·71-s + 390·73-s + 688·79-s + 1.18e3·83-s + 694·89-s + 1.72e3·97-s − 1.18e3·101-s − 1.96e3·103-s + 796·107-s + 342·109-s + 114·113-s + ⋯ |
| L(s) = 1 | − 0.109·11-s − 1.15·13-s + 1.62·17-s + 0.531·19-s + 0.870·23-s − 0.858·29-s − 1.57·31-s + 0.435·37-s + 0.0228·41-s − 0.0425·43-s − 0.620·47-s − 49-s + 1.69·53-s − 0.0794·59-s − 0.927·61-s + 0.342·67-s + 1.05·71-s + 0.625·73-s + 0.979·79-s + 1.57·83-s + 0.826·89-s + 1.80·97-s − 1.16·101-s − 1.88·103-s + 0.719·107-s + 0.300·109-s + 0.0949·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.953187726\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.953187726\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + 4 T + p^{3} T^{2} \) |
| 13 | \( 1 + 54 T + p^{3} T^{2} \) |
| 17 | \( 1 - 114 T + p^{3} T^{2} \) |
| 19 | \( 1 - 44 T + p^{3} T^{2} \) |
| 23 | \( 1 - 96 T + p^{3} T^{2} \) |
| 29 | \( 1 + 134 T + p^{3} T^{2} \) |
| 31 | \( 1 + 272 T + p^{3} T^{2} \) |
| 37 | \( 1 - 98 T + p^{3} T^{2} \) |
| 41 | \( 1 - 6 T + p^{3} T^{2} \) |
| 43 | \( 1 + 12 T + p^{3} T^{2} \) |
| 47 | \( 1 + 200 T + p^{3} T^{2} \) |
| 53 | \( 1 - 654 T + p^{3} T^{2} \) |
| 59 | \( 1 + 36 T + p^{3} T^{2} \) |
| 61 | \( 1 + 442 T + p^{3} T^{2} \) |
| 67 | \( 1 - 188 T + p^{3} T^{2} \) |
| 71 | \( 1 - 632 T + p^{3} T^{2} \) |
| 73 | \( 1 - 390 T + p^{3} T^{2} \) |
| 79 | \( 1 - 688 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1188 T + p^{3} T^{2} \) |
| 89 | \( 1 - 694 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1726 T + p^{3} T^{2} \) |
| 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
−9.120680461817596581941204715006, −7.86022129626494924319286886148, −7.52548048942281778584916924699, −6.61087202736489518576626168860, −5.43811594893332272319143307654, −5.09143335930864906597807117613, −3.79126924200260471936410035205, −3.00098549697199502841337192071, −1.86975000775240846554078848643, −0.65681238924601170841980941923,
0.65681238924601170841980941923, 1.86975000775240846554078848643, 3.00098549697199502841337192071, 3.79126924200260471936410035205, 5.09143335930864906597807117613, 5.43811594893332272319143307654, 6.61087202736489518576626168860, 7.52548048942281778584916924699, 7.86022129626494924319286886148, 9.120680461817596581941204715006
