| L(s) = 1 | − 3·5-s + 2·7-s − 11-s + 4·13-s − 6·17-s − 8·19-s + 3·23-s + 4·25-s + 5·31-s − 6·35-s + 37-s + 10·43-s − 3·49-s − 6·53-s + 3·55-s + 3·59-s + 4·61-s − 12·65-s + 67-s − 15·71-s − 4·73-s − 2·77-s + 2·79-s + 6·83-s + 18·85-s + 9·89-s + 8·91-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.755·7-s − 0.301·11-s + 1.10·13-s − 1.45·17-s − 1.83·19-s + 0.625·23-s + 4/5·25-s + 0.898·31-s − 1.01·35-s + 0.164·37-s + 1.52·43-s − 3/7·49-s − 0.824·53-s + 0.404·55-s + 0.390·59-s + 0.512·61-s − 1.48·65-s + 0.122·67-s − 1.78·71-s − 0.468·73-s − 0.227·77-s + 0.225·79-s + 0.658·83-s + 1.95·85-s + 0.953·89-s + 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.246818742\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.246818742\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−8.035390626139297528008844662539, −7.50615261503564981219001739421, −6.59522898242271340845554036308, −6.10061966836690236068984245763, −4.87872940749250314238046172195, −4.36218003026659083334803525471, −3.85052915187700989911233997211, −2.78805552633223734216042456786, −1.83610540162403363663667938387, −0.57447048966484511436229060871,
0.57447048966484511436229060871, 1.83610540162403363663667938387, 2.78805552633223734216042456786, 3.85052915187700989911233997211, 4.36218003026659083334803525471, 4.87872940749250314238046172195, 6.10061966836690236068984245763, 6.59522898242271340845554036308, 7.50615261503564981219001739421, 8.035390626139297528008844662539
