| L(s) = 1 | + 7-s − 11-s + 2·13-s + 2·17-s − 4·19-s − 6·29-s − 6·37-s + 6·41-s − 4·43-s + 49-s − 2·53-s + 4·59-s + 6·61-s + 12·67-s − 10·73-s − 77-s − 8·79-s + 4·83-s − 10·89-s + 2·91-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 2·119-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 0.301·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s − 1.11·29-s − 0.986·37-s + 0.937·41-s − 0.609·43-s + 1/7·49-s − 0.274·53-s + 0.520·59-s + 0.768·61-s + 1.46·67-s − 1.17·73-s − 0.113·77-s − 0.900·79-s + 0.439·83-s − 1.05·89-s + 0.209·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.183·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 277200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.887936664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.887936664\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 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 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.72315271673315, −12.44617624744205, −11.67644553491146, −11.45639338154133, −10.86127984704587, −10.57957163655525, −10.00000196671968, −9.574253037337527, −9.001411979299056, −8.475167271404716, −8.234000370420953, −7.626352768799258, −7.137020662798235, −6.688778780365374, −6.076899342571308, −5.569193519537254, −5.250609688409918, −4.535026029955891, −4.065352710560155, −3.565138953665256, −2.996427832628872, −2.277376680345245, −1.809143319080874, −1.173981737202725, −0.3791533462690881,
0.3791533462690881, 1.173981737202725, 1.809143319080874, 2.277376680345245, 2.996427832628872, 3.565138953665256, 4.065352710560155, 4.535026029955891, 5.250609688409918, 5.569193519537254, 6.076899342571308, 6.688778780365374, 7.137020662798235, 7.626352768799258, 8.234000370420953, 8.475167271404716, 9.001411979299056, 9.574253037337527, 10.00000196671968, 10.57957163655525, 10.86127984704587, 11.45639338154133, 11.67644553491146, 12.44617624744205, 12.72315271673315
