L(s) = 1 | + 5-s − 5·11-s − 13-s − 17-s − 4·19-s + 23-s + 25-s + 6·29-s + 6·31-s − 7·37-s − 7·41-s + 6·43-s − 10·47-s − 6·53-s − 5·55-s + 7·59-s − 2·61-s − 65-s − 67-s + 8·71-s − 73-s − 11·79-s + 6·83-s − 85-s + 15·89-s − 4·95-s − 7·97-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.50·11-s − 0.277·13-s − 0.242·17-s − 0.917·19-s + 0.208·23-s + 1/5·25-s + 1.11·29-s + 1.07·31-s − 1.15·37-s − 1.09·41-s + 0.914·43-s − 1.45·47-s − 0.824·53-s − 0.674·55-s + 0.911·59-s − 0.256·61-s − 0.124·65-s − 0.122·67-s + 0.949·71-s − 0.117·73-s − 1.23·79-s + 0.658·83-s − 0.108·85-s + 1.58·89-s − 0.410·95-s − 0.710·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 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
−13.17848706715213, −12.78579343774663, −12.24006066152160, −11.91481477283589, −11.16470956515041, −10.78367797885527, −10.34305170012423, −10.00173697120223, −9.560502241809817, −8.841323418871745, −8.406337726728855, −8.095238009731977, −7.548644413041202, −6.829412838404981, −6.571376608921510, −6.011192115612440, −5.309903533182563, −5.011418883379838, −4.552882635562069, −3.898829312942607, −3.040152527336644, −2.801363930044389, −2.124896420310321, −1.626243837748620, −0.6918348499557924, 0,
0.6918348499557924, 1.626243837748620, 2.124896420310321, 2.801363930044389, 3.040152527336644, 3.898829312942607, 4.552882635562069, 5.011418883379838, 5.309903533182563, 6.011192115612440, 6.571376608921510, 6.829412838404981, 7.548644413041202, 8.095238009731977, 8.406337726728855, 8.841323418871745, 9.560502241809817, 10.00173697120223, 10.34305170012423, 10.78367797885527, 11.16470956515041, 11.91481477283589, 12.24006066152160, 12.78579343774663, 13.17848706715213