| L(s) = 1 | − 3-s + 5-s − 4·7-s + 9-s − 15-s + 6·17-s − 4·19-s + 4·21-s + 25-s − 27-s − 6·29-s + 8·31-s − 4·35-s − 2·37-s + 6·41-s + 4·43-s + 45-s + 9·49-s − 6·51-s − 6·53-s + 4·57-s − 10·61-s − 4·63-s − 4·67-s − 2·73-s − 75-s − 8·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.258·15-s + 1.45·17-s − 0.917·19-s + 0.872·21-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.676·35-s − 0.328·37-s + 0.937·41-s + 0.609·43-s + 0.149·45-s + 9/7·49-s − 0.840·51-s − 0.824·53-s + 0.529·57-s − 1.28·61-s − 0.503·63-s − 0.488·67-s − 0.234·73-s − 0.115·75-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.15426734220360, −14.39761500143711, −14.01728964823858, −13.31246044115949, −12.85971497236781, −12.53939088653433, −12.02009976153553, −11.41443332346952, −10.68065158024567, −10.26092182680354, −9.858170800223634, −9.303571887510142, −8.873255664586473, −7.945712697270738, −7.489426866455043, −6.773803237738988, −6.224775058845279, −5.934684090660108, −5.339755316526572, −4.530361860064649, −3.896503439593459, −3.179542186420067, −2.678506450387527, −1.723027492017072, −0.8623820669315273, 0,
0.8623820669315273, 1.723027492017072, 2.678506450387527, 3.179542186420067, 3.896503439593459, 4.530361860064649, 5.339755316526572, 5.934684090660108, 6.224775058845279, 6.773803237738988, 7.489426866455043, 7.945712697270738, 8.873255664586473, 9.303571887510142, 9.858170800223634, 10.26092182680354, 10.68065158024567, 11.41443332346952, 12.02009976153553, 12.53939088653433, 12.85971497236781, 13.31246044115949, 14.01728964823858, 14.39761500143711, 15.15426734220360
