L(s) = 1 | − 5-s − 3·9-s + 4·11-s + 6·13-s − 6·17-s + 19-s + 8·23-s + 25-s + 2·29-s − 2·37-s + 2·41-s − 4·43-s + 3·45-s − 8·47-s − 7·49-s + 6·53-s − 4·55-s + 4·59-s + 2·61-s − 6·65-s − 8·67-s + 8·71-s + 2·73-s − 8·79-s + 9·81-s − 4·83-s + 6·85-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 9-s + 1.20·11-s + 1.66·13-s − 1.45·17-s + 0.229·19-s + 1.66·23-s + 1/5·25-s + 0.371·29-s − 0.328·37-s + 0.312·41-s − 0.609·43-s + 0.447·45-s − 1.16·47-s − 49-s + 0.824·53-s − 0.539·55-s + 0.520·59-s + 0.256·61-s − 0.744·65-s − 0.977·67-s + 0.949·71-s + 0.234·73-s − 0.900·79-s + 81-s − 0.439·83-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.839393981\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.839393981\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−8.361435156468919491742708969881, −7.24716202687168129413635330491, −6.55197436637470088547502867201, −6.15600151629972809033939840965, −5.15968148471021796410147652195, −4.38056019148119516932547433378, −3.56590809209375136513349937194, −2.99493262394815734872603724909, −1.74333899521904856200754209637, −0.73306037429571521476047301200,
0.73306037429571521476047301200, 1.74333899521904856200754209637, 2.99493262394815734872603724909, 3.56590809209375136513349937194, 4.38056019148119516932547433378, 5.15968148471021796410147652195, 6.15600151629972809033939840965, 6.55197436637470088547502867201, 7.24716202687168129413635330491, 8.361435156468919491742708969881