| L(s) = 1 | + 3·3-s + 5-s + 6·9-s − 6·13-s + 3·15-s + 2·17-s − 9·23-s + 25-s + 9·27-s − 3·29-s − 2·31-s + 8·37-s − 18·39-s + 5·41-s − 43-s + 6·45-s − 8·47-s + 6·51-s + 4·53-s + 8·59-s + 7·61-s − 6·65-s − 3·67-s − 27·69-s + 8·71-s + 14·73-s + 3·75-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s + 2·9-s − 1.66·13-s + 0.774·15-s + 0.485·17-s − 1.87·23-s + 1/5·25-s + 1.73·27-s − 0.557·29-s − 0.359·31-s + 1.31·37-s − 2.88·39-s + 0.780·41-s − 0.152·43-s + 0.894·45-s − 1.16·47-s + 0.840·51-s + 0.549·53-s + 1.04·59-s + 0.896·61-s − 0.744·65-s − 0.366·67-s − 3.25·69-s + 0.949·71-s + 1.63·73-s + 0.346·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 118580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 118580 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.00377327178778, −13.39698557134442, −12.91356427089226, −12.60088710781892, −12.00715749423389, −11.50008296117293, −10.75479229292836, −10.01040978502482, −9.797834828051653, −9.572024039982484, −8.970680452747617, −8.297745798421220, −7.944266822753547, −7.573929369435964, −7.032957248081221, −6.462364829330064, −5.710322057942796, −5.190875082405378, −4.481160273901427, −3.957683426412791, −3.493366136059161, −2.690599505661900, −2.328332794611145, −1.949490812906330, −1.098515948473071, 0,
1.098515948473071, 1.949490812906330, 2.328332794611145, 2.690599505661900, 3.493366136059161, 3.957683426412791, 4.481160273901427, 5.190875082405378, 5.710322057942796, 6.462364829330064, 7.032957248081221, 7.573929369435964, 7.944266822753547, 8.297745798421220, 8.970680452747617, 9.572024039982484, 9.797834828051653, 10.01040978502482, 10.75479229292836, 11.50008296117293, 12.00715749423389, 12.60088710781892, 12.91356427089226, 13.39698557134442, 14.00377327178778
