| L(s) = 1 | + 3-s − 2·4-s + 3·7-s + 9-s − 3·11-s − 2·12-s + 4·13-s + 4·16-s − 8·17-s − 6·19-s + 3·21-s − 3·23-s + 27-s − 6·28-s − 29-s + 4·31-s − 3·33-s − 2·36-s − 37-s + 4·39-s − 10·41-s + 8·43-s + 6·44-s + 47-s + 4·48-s + 2·49-s − 8·51-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 1.13·7-s + 1/3·9-s − 0.904·11-s − 0.577·12-s + 1.10·13-s + 16-s − 1.94·17-s − 1.37·19-s + 0.654·21-s − 0.625·23-s + 0.192·27-s − 1.13·28-s − 0.185·29-s + 0.718·31-s − 0.522·33-s − 1/3·36-s − 0.164·37-s + 0.640·39-s − 1.56·41-s + 1.21·43-s + 0.904·44-s + 0.145·47-s + 0.577·48-s + 2/7·49-s − 1.12·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−8.189160655810162844079283824971, −7.947156623676494102043124859035, −6.70113328797246898871580883519, −5.91855439893167809909963668026, −4.81556651532580505661343165060, −4.47002531004123040945885832995, −3.65346891646054397492655638360, −2.42999473402437639165431796003, −1.56481639580926788648226136614, 0,
1.56481639580926788648226136614, 2.42999473402437639165431796003, 3.65346891646054397492655638360, 4.47002531004123040945885832995, 4.81556651532580505661343165060, 5.91855439893167809909963668026, 6.70113328797246898871580883519, 7.947156623676494102043124859035, 8.189160655810162844079283824971
