| L(s) = 1 | + 3-s − 2·9-s − 5·11-s − 7·13-s − 3·17-s + 2·19-s + 8·23-s − 5·27-s + 5·29-s − 10·31-s − 5·33-s + 4·37-s − 7·39-s + 6·41-s − 2·43-s + 7·47-s − 3·51-s − 10·53-s + 2·57-s + 10·59-s − 12·61-s + 2·67-s + 8·69-s − 2·73-s + 7·79-s + 81-s + 4·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s − 1.50·11-s − 1.94·13-s − 0.727·17-s + 0.458·19-s + 1.66·23-s − 0.962·27-s + 0.928·29-s − 1.79·31-s − 0.870·33-s + 0.657·37-s − 1.12·39-s + 0.937·41-s − 0.304·43-s + 1.02·47-s − 0.420·51-s − 1.37·53-s + 0.264·57-s + 1.30·59-s − 1.53·61-s + 0.244·67-s + 0.963·69-s − 0.234·73-s + 0.787·79-s + 1/9·81-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−14.34900655177770, −13.76092755424015, −13.29364555132892, −12.68264278154094, −12.56795986191701, −11.73253644024035, −11.24240071152713, −10.73661763159155, −10.30994996195709, −9.628360375926139, −9.110198406386657, −8.914113739195957, −7.989588977713458, −7.686466299103204, −7.293904258912949, −6.688892960767231, −5.860937193979010, −5.277411222848195, −4.926699649334270, −4.409940184730153, −3.372038417536561, −2.935582346166269, −2.425392138180120, −2.022659438820859, −0.7294860793202408, 0,
0.7294860793202408, 2.022659438820859, 2.425392138180120, 2.935582346166269, 3.372038417536561, 4.409940184730153, 4.926699649334270, 5.277411222848195, 5.860937193979010, 6.688892960767231, 7.293904258912949, 7.686466299103204, 7.989588977713458, 8.914113739195957, 9.110198406386657, 9.628360375926139, 10.30994996195709, 10.73661763159155, 11.24240071152713, 11.73253644024035, 12.56795986191701, 12.68264278154094, 13.29364555132892, 13.76092755424015, 14.34900655177770
