| L(s) = 1 | − 5.27·7-s + 6.31·13-s − 3.46·19-s − 5·25-s − 4.52·31-s + 11.9·37-s + 10.3·43-s + 20.8·49-s + 5.00·61-s − 16·67-s − 13.8·73-s + 15.0·79-s − 33.3·91-s − 13.8·97-s − 16.5·103-s + 13.2·109-s + ⋯ |
| L(s) = 1 | − 1.99·7-s + 1.75·13-s − 0.794·19-s − 25-s − 0.811·31-s + 1.96·37-s + 1.58·43-s + 2.97·49-s + 0.640·61-s − 1.95·67-s − 1.62·73-s + 1.69·79-s − 3.49·91-s − 1.40·97-s − 1.63·103-s + 1.27·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 5.27T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 6.31T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4.52T + 31T^{2} \) |
| 37 | \( 1 - 11.9T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 5.00T + 61T^{2} \) |
| 67 | \( 1 + 16T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−7.29714853558947201775910760975, −6.52914055935490243195710933975, −5.99284720554359555506112420321, −5.75083672563839655216111551405, −4.25160323299228911034355326772, −3.85631421440277869096500323868, −3.11958349431956666006985334246, −2.34404997446115660914470319164, −1.08862081458042372228834389055, 0,
1.08862081458042372228834389055, 2.34404997446115660914470319164, 3.11958349431956666006985334246, 3.85631421440277869096500323868, 4.25160323299228911034355326772, 5.75083672563839655216111551405, 5.99284720554359555506112420321, 6.52914055935490243195710933975, 7.29714853558947201775910760975
