L(s) = 1 | − 3-s − 3·5-s + 7-s + 9-s + 11-s − 13-s + 3·15-s − 17-s + 6·19-s − 21-s + 2·23-s + 4·25-s − 27-s + 2·29-s − 33-s − 3·35-s − 5·37-s + 39-s + 4·41-s − 9·43-s − 3·45-s + 49-s + 51-s − 11·53-s − 3·55-s − 6·57-s + 4·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.34·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s + 0.774·15-s − 0.242·17-s + 1.37·19-s − 0.218·21-s + 0.417·23-s + 4/5·25-s − 0.192·27-s + 0.371·29-s − 0.174·33-s − 0.507·35-s − 0.821·37-s + 0.160·39-s + 0.624·41-s − 1.37·43-s − 0.447·45-s + 1/7·49-s + 0.140·51-s − 1.51·53-s − 0.404·55-s − 0.794·57-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−15.60981613941765, −15.45130075832274, −14.76209349314582, −14.20758232848206, −13.60285876341698, −12.96457627358595, −12.19488031716986, −11.94362431453269, −11.53787877650998, −10.90446940910913, −10.51910697977352, −9.608930914063482, −9.211499601922779, −8.330834776557479, −7.916968657437016, −7.329842444905165, −6.839678661816177, −6.165098898053451, −5.231997407026314, −4.887468730696072, −4.189691837216708, −3.514277575748094, −2.926803019119002, −1.747222139528291, −0.9146719091179843, 0,
0.9146719091179843, 1.747222139528291, 2.926803019119002, 3.514277575748094, 4.189691837216708, 4.887468730696072, 5.231997407026314, 6.165098898053451, 6.839678661816177, 7.329842444905165, 7.916968657437016, 8.330834776557479, 9.211499601922779, 9.608930914063482, 10.51910697977352, 10.90446940910913, 11.53787877650998, 11.94362431453269, 12.19488031716986, 12.96457627358595, 13.60285876341698, 14.20758232848206, 14.76209349314582, 15.45130075832274, 15.60981613941765