| L(s) = 1 | − 7-s − 2·17-s + 4·19-s + 4·23-s − 5·25-s + 6·29-s − 4·31-s + 6·37-s − 6·41-s + 4·43-s + 12·47-s + 49-s + 4·53-s + 8·61-s − 12·67-s − 4·71-s − 4·73-s − 4·83-s + 12·89-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 2·119-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 0.485·17-s + 0.917·19-s + 0.834·23-s − 25-s + 1.11·29-s − 0.718·31-s + 0.986·37-s − 0.937·41-s + 0.609·43-s + 1.75·47-s + 1/7·49-s + 0.549·53-s + 1.02·61-s − 1.46·67-s − 0.474·71-s − 0.468·73-s − 0.439·83-s + 1.27·89-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.183·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−13.64780342605733, −13.36519567492561, −12.92973616131402, −12.26886396105563, −11.84081297663825, −11.51979036522965, −10.75288036691950, −10.49934038487087, −9.855578397711499, −9.373833185207172, −8.984098307271877, −8.441183296725346, −7.820114653211478, −7.300828540526413, −6.921043248418536, −6.281060706443067, −5.733635761444715, −5.304163230898281, −4.614959072428674, −4.056089698185712, −3.542775526487244, −2.761608716011342, −2.455281245169985, −1.487461773878241, −0.8817954821509717, 0,
0.8817954821509717, 1.487461773878241, 2.455281245169985, 2.761608716011342, 3.542775526487244, 4.056089698185712, 4.614959072428674, 5.304163230898281, 5.733635761444715, 6.281060706443067, 6.921043248418536, 7.300828540526413, 7.820114653211478, 8.441183296725346, 8.984098307271877, 9.373833185207172, 9.855578397711499, 10.49934038487087, 10.75288036691950, 11.51979036522965, 11.84081297663825, 12.26886396105563, 12.92973616131402, 13.36519567492561, 13.64780342605733
