L(s) = 1 | − 0.378·7-s − 3.48·13-s + 3.46·19-s − 5·25-s + 10.1·31-s + 2.17·37-s − 10.3·43-s − 6.85·49-s + 14.7·61-s − 16·67-s + 13.8·73-s − 9.41·79-s + 1.32·91-s + 13.8·97-s − 11.6·103-s − 16.1·109-s + ⋯ |
L(s) = 1 | − 0.143·7-s − 0.966·13-s + 0.794·19-s − 25-s + 1.82·31-s + 0.357·37-s − 1.58·43-s − 0.979·49-s + 1.89·61-s − 1.95·67-s + 1.62·73-s − 1.05·79-s + 0.138·91-s + 1.40·97-s − 1.15·103-s − 1.54·109-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 | \( 1 + 0.378T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 3.48T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 - 2.17T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 14.7T + 61T^{2} \) |
| 67 | \( 1 + 16T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 + 9.41T + 79T^{2} \) |
| 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
−7.40632572199544475748006133165, −6.69829297034376627491740885298, −6.07469074626443437791982231516, −5.21114722170712189196749862302, −4.69973857411545586297705161694, −3.81538375692025753991748620165, −2.99891639345788159405111319893, −2.27196005744173555007763847150, −1.21377998738872618041691397987, 0,
1.21377998738872618041691397987, 2.27196005744173555007763847150, 2.99891639345788159405111319893, 3.81538375692025753991748620165, 4.69973857411545586297705161694, 5.21114722170712189196749862302, 6.07469074626443437791982231516, 6.69829297034376627491740885298, 7.40632572199544475748006133165