| L(s) = 1 | + 3-s − 5-s − 3·7-s + 9-s − 2·11-s + 3·13-s − 15-s + 6·17-s + 4·19-s − 3·21-s + 25-s + 27-s − 4·29-s + 3·31-s − 2·33-s + 3·35-s + 3·39-s + 8·41-s − 43-s − 45-s + 2·49-s + 6·51-s + 8·53-s + 2·55-s + 4·57-s − 14·59-s + 10·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.13·7-s + 1/3·9-s − 0.603·11-s + 0.832·13-s − 0.258·15-s + 1.45·17-s + 0.917·19-s − 0.654·21-s + 1/5·25-s + 0.192·27-s − 0.742·29-s + 0.538·31-s − 0.348·33-s + 0.507·35-s + 0.480·39-s + 1.24·41-s − 0.152·43-s − 0.149·45-s + 2/7·49-s + 0.840·51-s + 1.09·53-s + 0.269·55-s + 0.529·57-s − 1.82·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.776168781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.776168781\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
| 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
−12.65037351583107, −12.30269047310382, −11.66611917474708, −11.33332611826241, −10.68827695075087, −10.23448231292593, −9.845868704197929, −9.439714287936097, −8.992307603017230, −8.428566196941016, −7.980849569907796, −7.465696951332631, −7.295967528296230, −6.481183437903156, −6.144024573973132, −5.483928275390367, −5.211016549615146, −4.352140345004521, −3.791534210388196, −3.505413884190173, −2.887419337052522, −2.663676586272854, −1.688015500637688, −1.052426864531253, −0.4740897468519829,
0.4740897468519829, 1.052426864531253, 1.688015500637688, 2.663676586272854, 2.887419337052522, 3.505413884190173, 3.791534210388196, 4.352140345004521, 5.211016549615146, 5.483928275390367, 6.144024573973132, 6.481183437903156, 7.295967528296230, 7.465696951332631, 7.980849569907796, 8.428566196941016, 8.992307603017230, 9.439714287936097, 9.845868704197929, 10.23448231292593, 10.68827695075087, 11.33332611826241, 11.66611917474708, 12.30269047310382, 12.65037351583107
