L(s) = 1 | − 5-s − 4·7-s − 11-s − 13-s + 17-s + 19-s − 6·23-s + 25-s − 8·29-s + 2·31-s + 4·35-s + 5·37-s + 5·41-s − 5·43-s + 9·47-s + 9·49-s + 55-s + 6·59-s − 2·61-s + 65-s + 3·67-s + 10·71-s + 16·73-s + 4·77-s − 14·79-s + 2·83-s − 85-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 1.51·7-s − 0.301·11-s − 0.277·13-s + 0.242·17-s + 0.229·19-s − 1.25·23-s + 1/5·25-s − 1.48·29-s + 0.359·31-s + 0.676·35-s + 0.821·37-s + 0.780·41-s − 0.762·43-s + 1.31·47-s + 9/7·49-s + 0.134·55-s + 0.781·59-s − 0.256·61-s + 0.124·65-s + 0.366·67-s + 1.18·71-s + 1.87·73-s + 0.455·77-s − 1.57·79-s + 0.219·83-s − 0.108·85-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
\(L(1)\) |
\(\approx\) |
\(0.6523950353\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6523950353\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−13.58394150116102, −13.26030968799960, −12.56615029873638, −12.48976265109341, −11.82769100701042, −11.30494046880458, −10.79186960547406, −10.15631389311420, −9.751859144317212, −9.429187274403771, −8.852492704479313, −8.100227873313385, −7.781741205923418, −7.143587178158525, −6.722365219632868, −6.082744849595022, −5.667448864308764, −5.111523067135272, −4.191862714620916, −3.865598435176884, −3.318348899458966, −2.624572757349419, −2.168806414926272, −1.094324603426468, −0.2762053371312800,
0.2762053371312800, 1.094324603426468, 2.168806414926272, 2.624572757349419, 3.318348899458966, 3.865598435176884, 4.191862714620916, 5.111523067135272, 5.667448864308764, 6.082744849595022, 6.722365219632868, 7.143587178158525, 7.781741205923418, 8.100227873313385, 8.852492704479313, 9.429187274403771, 9.751859144317212, 10.15631389311420, 10.79186960547406, 11.30494046880458, 11.82769100701042, 12.48976265109341, 12.56615029873638, 13.26030968799960, 13.58394150116102