L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 2·6-s + 9-s − 2·11-s − 2·12-s − 13-s − 4·16-s − 2·18-s + 19-s + 4·22-s + 2·26-s − 27-s + 4·29-s + 9·31-s + 8·32-s + 2·33-s + 2·36-s − 3·37-s − 2·38-s + 39-s − 10·41-s − 5·43-s − 4·44-s + 6·47-s + 4·48-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.816·6-s + 1/3·9-s − 0.603·11-s − 0.577·12-s − 0.277·13-s − 16-s − 0.471·18-s + 0.229·19-s + 0.852·22-s + 0.392·26-s − 0.192·27-s + 0.742·29-s + 1.61·31-s + 1.41·32-s + 0.348·33-s + 1/3·36-s − 0.493·37-s − 0.324·38-s + 0.160·39-s − 1.56·41-s − 0.762·43-s − 0.603·44-s + 0.875·47-s + 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−8.212220448758386434880997358484, −7.62041984330376402307537675857, −6.83132953118780910312225320263, −6.22804484524633661645852904252, −5.09065507826252211926217580230, −4.56907831712214781328451733869, −3.21420965360206411348140268323, −2.13842260198962939247232737734, −1.08983332946246077622289261182, 0,
1.08983332946246077622289261182, 2.13842260198962939247232737734, 3.21420965360206411348140268323, 4.56907831712214781328451733869, 5.09065507826252211926217580230, 6.22804484524633661645852904252, 6.83132953118780910312225320263, 7.62041984330376402307537675857, 8.212220448758386434880997358484