| L(s) = 1 | − 3-s − 2.82·5-s + 4.24·7-s + 9-s + 4·11-s − 4.24·13-s + 2.82·15-s + 6·17-s + 2·19-s − 4.24·21-s + 2.82·23-s + 3.00·25-s − 27-s + 5.65·29-s − 4.24·31-s − 4·33-s − 12·35-s − 4.24·37-s + 4.24·39-s − 10·41-s − 6·43-s − 2.82·45-s − 2.82·47-s + 10.9·49-s − 6·51-s + 5.65·53-s − 11.3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.26·5-s + 1.60·7-s + 0.333·9-s + 1.20·11-s − 1.17·13-s + 0.730·15-s + 1.45·17-s + 0.458·19-s − 0.925·21-s + 0.589·23-s + 0.600·25-s − 0.192·27-s + 1.05·29-s − 0.762·31-s − 0.696·33-s − 2.02·35-s − 0.697·37-s + 0.679·39-s − 1.56·41-s − 0.914·43-s − 0.421·45-s − 0.412·47-s + 1.57·49-s − 0.840·51-s + 0.777·53-s − 1.52·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.524545247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.524545247\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 5.65T + 29T^{2} \) |
| 31 | \( 1 + 4.24T + 31T^{2} \) |
| 37 | \( 1 + 4.24T + 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 5.65T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 4.24T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 - 16T + 73T^{2} \) |
| 79 | \( 1 + 4.24T + 79T^{2} \) |
| 83 | \( 1 - 16T + 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
| 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
−8.435958671936059435057492751632, −7.954005617676551178288923135094, −7.25895623666010637445567580839, −6.67806811338403373458637637059, −5.22363521130256737612579690330, −5.04161423499886678986819300446, −4.06887742984739526411385386008, −3.32815351118757725108372328974, −1.78983407007564537238033898748, −0.819378412879548951261293557368,
0.819378412879548951261293557368, 1.78983407007564537238033898748, 3.32815351118757725108372328974, 4.06887742984739526411385386008, 5.04161423499886678986819300446, 5.22363521130256737612579690330, 6.67806811338403373458637637059, 7.25895623666010637445567580839, 7.954005617676551178288923135094, 8.435958671936059435057492751632
