L(s) = 1 | − 5-s + 4·7-s + 4·11-s − 2·13-s − 2·17-s − 4·19-s + 4·23-s + 25-s + 2·29-s + 8·31-s − 4·35-s + 6·37-s + 6·41-s + 8·43-s + 4·47-s + 9·49-s − 6·53-s − 4·55-s − 4·59-s − 2·61-s + 2·65-s − 8·67-s − 6·73-s + 16·77-s − 16·83-s + 2·85-s + 6·89-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.51·7-s + 1.20·11-s − 0.554·13-s − 0.485·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.676·35-s + 0.986·37-s + 0.937·41-s + 1.21·43-s + 0.583·47-s + 9/7·49-s − 0.824·53-s − 0.539·55-s − 0.520·59-s − 0.256·61-s + 0.248·65-s − 0.977·67-s − 0.702·73-s + 1.82·77-s − 1.75·83-s + 0.216·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.714051882\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.714051882\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−10.65133305239162895186920528545, −9.401949169928688362704456671527, −8.635268153066941217007271839152, −7.88125929411027787932176182764, −7.00520502565783528594752431202, −5.99573481729811170443753568761, −4.61878022728753276110930411726, −4.28002313553101117805216295071, −2.60538869357449517223391665122, −1.23519091396578725398334381678,
1.23519091396578725398334381678, 2.60538869357449517223391665122, 4.28002313553101117805216295071, 4.61878022728753276110930411726, 5.99573481729811170443753568761, 7.00520502565783528594752431202, 7.88125929411027787932176182764, 8.635268153066941217007271839152, 9.401949169928688362704456671527, 10.65133305239162895186920528545