| L(s) = 1 | + 3·3-s + 5-s − 7-s + 6·9-s + 5·11-s − 5·13-s + 3·15-s − 7·17-s + 2·19-s − 3·21-s + 2·23-s + 25-s + 9·27-s + 7·29-s − 4·31-s + 15·33-s − 35-s − 6·37-s − 15·39-s − 12·41-s + 2·43-s + 6·45-s − 47-s + 49-s − 21·51-s + 5·55-s + 6·57-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s − 0.377·7-s + 2·9-s + 1.50·11-s − 1.38·13-s + 0.774·15-s − 1.69·17-s + 0.458·19-s − 0.654·21-s + 0.417·23-s + 1/5·25-s + 1.73·27-s + 1.29·29-s − 0.718·31-s + 2.61·33-s − 0.169·35-s − 0.986·37-s − 2.40·39-s − 1.87·41-s + 0.304·43-s + 0.894·45-s − 0.145·47-s + 1/7·49-s − 2.94·51-s + 0.674·55-s + 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.643691978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.643691978\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−10.40941159615388893250715508557, −9.543008020385817793332482027366, −9.086197799545514749819743984163, −8.375269634382673437491035425421, −7.07941858965721271749848204581, −6.67578280443386247224047726882, −4.88153619134139118802581016378, −3.84750334932192632607855993535, −2.79102468168229631071339860625, −1.80180608530670977304934504851,
1.80180608530670977304934504851, 2.79102468168229631071339860625, 3.84750334932192632607855993535, 4.88153619134139118802581016378, 6.67578280443386247224047726882, 7.07941858965721271749848204581, 8.375269634382673437491035425421, 9.086197799545514749819743984163, 9.543008020385817793332482027366, 10.40941159615388893250715508557
