L(s) = 1 | + 2·5-s − 7-s − 4·11-s − 2·17-s − 4·19-s + 4·23-s − 25-s − 10·29-s − 8·31-s − 2·35-s − 2·37-s + 10·41-s + 4·43-s − 8·47-s + 49-s + 6·53-s − 8·55-s − 8·59-s + 2·61-s + 8·67-s + 8·71-s − 6·73-s + 4·77-s + 8·79-s + 8·83-s − 4·85-s + 2·89-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 0.377·7-s − 1.20·11-s − 0.485·17-s − 0.917·19-s + 0.834·23-s − 1/5·25-s − 1.85·29-s − 1.43·31-s − 0.338·35-s − 0.328·37-s + 1.56·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s + 0.824·53-s − 1.07·55-s − 1.04·59-s + 0.256·61-s + 0.977·67-s + 0.949·71-s − 0.702·73-s + 0.455·77-s + 0.900·79-s + 0.878·83-s − 0.433·85-s + 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9865260177\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9865260177\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.71390604377042, −13.37659720082792, −12.95920039709914, −12.69820822408854, −12.10344255837002, −11.12715396737406, −10.91669157116235, −10.63016110278940, −9.789056159740779, −9.427534800340824, −9.110132166047618, −8.384573073331022, −7.826839923701884, −7.285158013714507, −6.779661397484217, −6.131207954390201, −5.583734187577589, −5.325247615244531, −4.568863398466889, −3.871052500460852, −3.313471544826586, −2.361049921949541, −2.256051770765967, −1.412419977401285, −0.3026326179423041,
0.3026326179423041, 1.412419977401285, 2.256051770765967, 2.361049921949541, 3.313471544826586, 3.871052500460852, 4.568863398466889, 5.325247615244531, 5.583734187577589, 6.131207954390201, 6.779661397484217, 7.285158013714507, 7.826839923701884, 8.384573073331022, 9.110132166047618, 9.427534800340824, 9.789056159740779, 10.63016110278940, 10.91669157116235, 11.12715396737406, 12.10344255837002, 12.69820822408854, 12.95920039709914, 13.37659720082792, 13.71390604377042