L(s) = 1 | − 5-s − 4·7-s − 11-s + 13-s + 2·17-s − 4·23-s + 25-s − 10·29-s + 4·35-s + 6·37-s − 10·41-s − 4·43-s + 9·49-s + 10·53-s + 55-s + 4·59-s − 10·61-s − 65-s − 12·67-s − 8·71-s − 2·73-s + 4·77-s − 4·83-s − 2·85-s + 6·89-s − 4·91-s + 6·97-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 1.51·7-s − 0.301·11-s + 0.277·13-s + 0.485·17-s − 0.834·23-s + 1/5·25-s − 1.85·29-s + 0.676·35-s + 0.986·37-s − 1.56·41-s − 0.609·43-s + 9/7·49-s + 1.37·53-s + 0.134·55-s + 0.520·59-s − 1.28·61-s − 0.124·65-s − 1.46·67-s − 0.949·71-s − 0.234·73-s + 0.455·77-s − 0.439·83-s − 0.216·85-s + 0.635·89-s − 0.419·91-s + 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1904381534\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1904381534\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.53046957623745, −13.14304085346084, −12.96318047288702, −12.13988892964045, −11.91288798420587, −11.38338902141098, −10.64279206363741, −10.27894829632377, −9.809985304261627, −9.305208451693449, −8.847390588221722, −8.226794126203655, −7.653482588789030, −7.216252700242427, −6.695264112466122, −5.998790395047531, −5.787585871719732, −5.054373483401889, −4.286182203849061, −3.729415781200526, −3.339303000456915, −2.766554876330780, −2.006962968191166, −1.197817839801481, −0.1413527167990019,
0.1413527167990019, 1.197817839801481, 2.006962968191166, 2.766554876330780, 3.339303000456915, 3.729415781200526, 4.286182203849061, 5.054373483401889, 5.787585871719732, 5.998790395047531, 6.695264112466122, 7.216252700242427, 7.653482588789030, 8.226794126203655, 8.847390588221722, 9.305208451693449, 9.809985304261627, 10.27894829632377, 10.64279206363741, 11.38338902141098, 11.91288798420587, 12.13988892964045, 12.96318047288702, 13.14304085346084, 13.53046957623745