L(s) = 1 | − 3-s + 5-s + 7-s + 9-s + 4·11-s + 6·13-s − 15-s + 6·17-s − 21-s + 4·23-s + 25-s − 27-s + 6·29-s − 4·33-s + 35-s − 2·37-s − 6·39-s + 2·41-s + 4·43-s + 45-s − 4·47-s + 49-s − 6·51-s + 6·53-s + 4·55-s + 12·59-s + 10·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 1.66·13-s − 0.258·15-s + 1.45·17-s − 0.218·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.696·33-s + 0.169·35-s − 0.328·37-s − 0.960·39-s + 0.312·41-s + 0.609·43-s + 0.149·45-s − 0.583·47-s + 1/7·49-s − 0.840·51-s + 0.824·53-s + 0.539·55-s + 1.56·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.743822811\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.743822811\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 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 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−8.091814960945815776456264607969, −7.02618680094950128125366166463, −6.60524464671489745849137447263, −5.74360707289160231244110082463, −5.41982112131558398553821853222, −4.31091734830001177834627960398, −3.73545395675063132271356615124, −2.78961351737865894884063981659, −1.34596882192917408053705998908, −1.11028961898316802987075858482,
1.11028961898316802987075858482, 1.34596882192917408053705998908, 2.78961351737865894884063981659, 3.73545395675063132271356615124, 4.31091734830001177834627960398, 5.41982112131558398553821853222, 5.74360707289160231244110082463, 6.60524464671489745849137447263, 7.02618680094950128125366166463, 8.091814960945815776456264607969