| L(s) = 1 | + 7-s − 2·11-s − 4·13-s + 19-s + 6·23-s − 5·25-s − 2·29-s + 4·31-s + 2·37-s − 2·41-s − 8·43-s + 8·47-s + 49-s + 6·53-s + 4·59-s + 14·61-s − 2·67-s − 6·73-s − 2·77-s − 10·79-s − 18·89-s − 4·91-s − 16·97-s − 16·101-s − 4·103-s + 2·109-s − 2·113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 0.603·11-s − 1.10·13-s + 0.229·19-s + 1.25·23-s − 25-s − 0.371·29-s + 0.718·31-s + 0.328·37-s − 0.312·41-s − 1.21·43-s + 1.16·47-s + 1/7·49-s + 0.824·53-s + 0.520·59-s + 1.79·61-s − 0.244·67-s − 0.702·73-s − 0.227·77-s − 1.12·79-s − 1.90·89-s − 0.419·91-s − 1.62·97-s − 1.59·101-s − 0.394·103-s + 0.191·109-s − 0.188·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−7.12425823460731832163837763563, −7.03280257949081827866268529434, −5.78117347559011741150560265627, −5.32199175865488799669057046658, −4.63968257154253098389230167267, −3.88863314822740631229449908794, −2.86570337240022274398725752601, −2.30676489624627270638819939507, −1.21908362361102348228728486834, 0,
1.21908362361102348228728486834, 2.30676489624627270638819939507, 2.86570337240022274398725752601, 3.88863314822740631229449908794, 4.63968257154253098389230167267, 5.32199175865488799669057046658, 5.78117347559011741150560265627, 7.03280257949081827866268529434, 7.12425823460731832163837763563
