| L(s) = 1 | + 3-s + 9-s + 13-s − 2·17-s − 2·19-s + 8·23-s − 5·25-s + 27-s − 6·29-s − 2·31-s + 6·37-s + 39-s − 4·43-s − 8·47-s − 2·51-s + 6·53-s − 2·57-s + 4·59-s + 2·61-s − 2·67-s + 8·69-s + 4·71-s + 2·73-s − 5·75-s + 12·79-s + 81-s − 12·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.277·13-s − 0.485·17-s − 0.458·19-s + 1.66·23-s − 25-s + 0.192·27-s − 1.11·29-s − 0.359·31-s + 0.986·37-s + 0.160·39-s − 0.609·43-s − 1.16·47-s − 0.280·51-s + 0.824·53-s − 0.264·57-s + 0.520·59-s + 0.256·61-s − 0.244·67-s + 0.963·69-s + 0.474·71-s + 0.234·73-s − 0.577·75-s + 1.35·79-s + 1/9·81-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.52562195780401, −13.44543212118097, −12.85581775855816, −12.60419012105094, −11.68355057181981, −11.40939558780758, −10.91998187672108, −10.41338585629584, −9.750285634583861, −9.380388195481746, −8.934911597198142, −8.361661030732470, −7.989008547379195, −7.317551787227975, −6.923979642298097, −6.380236015398276, −5.737601727913129, −5.195812160033450, −4.594238137002492, −3.988269989782356, −3.520231771909126, −2.893242336730792, −2.241672097153826, −1.704367391317378, −0.9262781939166756, 0,
0.9262781939166756, 1.704367391317378, 2.241672097153826, 2.893242336730792, 3.520231771909126, 3.988269989782356, 4.594238137002492, 5.195812160033450, 5.737601727913129, 6.380236015398276, 6.923979642298097, 7.317551787227975, 7.989008547379195, 8.361661030732470, 8.934911597198142, 9.380388195481746, 9.750285634583861, 10.41338585629584, 10.91998187672108, 11.40939558780758, 11.68355057181981, 12.60419012105094, 12.85581775855816, 13.44543212118097, 13.52562195780401
