| L(s) = 1 | − 4·7-s − 13-s + 3·19-s − 4·23-s + 29-s + 8·31-s − 37-s − 41-s − 6·43-s − 11·47-s + 9·49-s + 3·53-s + 10·59-s + 4·61-s − 13·67-s + 9·71-s + 3·79-s + 2·83-s − 10·89-s + 4·91-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.277·13-s + 0.688·19-s − 0.834·23-s + 0.185·29-s + 1.43·31-s − 0.164·37-s − 0.156·41-s − 0.914·43-s − 1.60·47-s + 9/7·49-s + 0.412·53-s + 1.30·59-s + 0.512·61-s − 1.58·67-s + 1.06·71-s + 0.337·79-s + 0.219·83-s − 1.05·89-s + 0.419·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.86738438127666, −13.57210231205710, −13.14960436114155, −12.59904316248236, −12.10607764602061, −11.74525100669485, −11.20324058391140, −10.39765245808319, −9.947393652321547, −9.813013841591284, −9.221538803161203, −8.506345311951324, −8.154280840496307, −7.440554178117634, −6.879390451733836, −6.462246065226467, −6.035598568500376, −5.345649177910461, −4.818344758281670, −4.069858862504945, −3.523131488627213, −2.995068509638217, −2.480226751327607, −1.633163923865516, −0.7419798199944283, 0,
0.7419798199944283, 1.633163923865516, 2.480226751327607, 2.995068509638217, 3.523131488627213, 4.069858862504945, 4.818344758281670, 5.345649177910461, 6.035598568500376, 6.462246065226467, 6.879390451733836, 7.440554178117634, 8.154280840496307, 8.506345311951324, 9.221538803161203, 9.813013841591284, 9.947393652321547, 10.39765245808319, 11.20324058391140, 11.74525100669485, 12.10607764602061, 12.59904316248236, 13.14960436114155, 13.57210231205710, 13.86738438127666
