| L(s) = 1 | + 3-s + 9-s − 4·11-s − 6·13-s − 2·17-s + 8·19-s − 4·23-s + 27-s + 6·29-s − 4·31-s − 4·33-s + 2·37-s − 6·39-s + 2·41-s + 12·43-s − 2·51-s − 2·53-s + 8·57-s + 4·59-s − 6·61-s + 4·67-s − 4·69-s + 8·71-s + 6·73-s − 16·79-s + 81-s − 4·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.20·11-s − 1.66·13-s − 0.485·17-s + 1.83·19-s − 0.834·23-s + 0.192·27-s + 1.11·29-s − 0.718·31-s − 0.696·33-s + 0.328·37-s − 0.960·39-s + 0.312·41-s + 1.82·43-s − 0.280·51-s − 0.274·53-s + 1.05·57-s + 0.520·59-s − 0.768·61-s + 0.488·67-s − 0.481·69-s + 0.949·71-s + 0.702·73-s − 1.80·79-s + 1/9·81-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−15.54058652133984, −14.74679436185680, −14.42515857271927, −13.80387713810318, −13.49263792422249, −12.61880299927393, −12.42261150759250, −11.79357919300899, −11.09949654118830, −10.50375238981424, −9.877809818028025, −9.594049821392796, −8.994880840667898, −8.186491615758924, −7.664888791499638, −7.412410664494057, −6.732483025188338, −5.808096558353727, −5.259197191432611, −4.734744303053425, −4.058881587611031, −3.159602454999920, −2.593351510733988, −2.171897924706484, −1.016361134923733, 0,
1.016361134923733, 2.171897924706484, 2.593351510733988, 3.159602454999920, 4.058881587611031, 4.734744303053425, 5.259197191432611, 5.808096558353727, 6.732483025188338, 7.412410664494057, 7.664888791499638, 8.186491615758924, 8.994880840667898, 9.594049821392796, 9.877809818028025, 10.50375238981424, 11.09949654118830, 11.79357919300899, 12.42261150759250, 12.61880299927393, 13.49263792422249, 13.80387713810318, 14.42515857271927, 14.74679436185680, 15.54058652133984
