| L(s) = 1 | + 4-s − 7-s + 11-s + 16-s + 17-s − 19-s + 25-s − 28-s − 37-s + 41-s + 44-s + 53-s + 64-s + 68-s + 71-s − 76-s − 77-s + 2·79-s − 2·83-s − 2·89-s + 2·97-s + 100-s + 101-s − 103-s − 2·107-s − 109-s − 112-s + ⋯ |
| L(s) = 1 | + 4-s − 7-s + 11-s + 16-s + 17-s − 19-s + 25-s − 28-s − 37-s + 41-s + 44-s + 53-s + 64-s + 68-s + 71-s − 76-s − 77-s + 2·79-s − 2·83-s − 2·89-s + 2·97-s + 100-s + 101-s − 103-s − 2·107-s − 109-s − 112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2763 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2763 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.526666034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.526666034\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 307 | \( 1 - T \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )^{2} \) |
| 83 | \( ( 1 + T )^{2} \) |
| 89 | \( ( 1 + T )^{2} \) |
| 97 | \( ( 1 - T )^{2} \) |
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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
−9.031496250857568065587155304921, −8.230691774644865006571559350651, −7.27509353171441064668529665897, −6.67174892603536315401250055181, −6.17370546681474951641750470565, −5.30741109745834586200319018097, −4.01880555798556468503300394296, −3.30612738242493762106617921067, −2.42732851850951898444877595082, −1.23052693357635021923380690542,
1.23052693357635021923380690542, 2.42732851850951898444877595082, 3.30612738242493762106617921067, 4.01880555798556468503300394296, 5.30741109745834586200319018097, 6.17370546681474951641750470565, 6.67174892603536315401250055181, 7.27509353171441064668529665897, 8.230691774644865006571559350651, 9.031496250857568065587155304921
