| L(s) = 1 | + 3-s + 2·5-s − 7-s + 9-s + 6·13-s + 2·15-s + 17-s − 21-s − 8·23-s − 25-s + 27-s + 6·29-s − 8·31-s − 2·35-s − 10·37-s + 6·39-s − 6·41-s − 12·43-s + 2·45-s + 49-s + 51-s + 10·53-s + 8·59-s − 6·61-s − 63-s + 12·65-s − 12·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.66·13-s + 0.516·15-s + 0.242·17-s − 0.218·21-s − 1.66·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.338·35-s − 1.64·37-s + 0.960·39-s − 0.937·41-s − 1.82·43-s + 0.298·45-s + 1/7·49-s + 0.140·51-s + 1.37·53-s + 1.04·59-s − 0.768·61-s − 0.125·63-s + 1.48·65-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−15.79143997149466, −15.27122220415730, −14.53598319707156, −14.01429978014087, −13.57949287274309, −13.31171001104949, −12.66321593643214, −11.89143080134049, −11.55450228143933, −10.44757105212995, −10.31542397638784, −9.802754342701026, −8.963891006076287, −8.621460233465600, −8.154376577888664, −7.258165403246340, −6.703235744523883, −5.999580912592519, −5.690550836835603, −4.844200458800341, −3.815738986514928, −3.590404087911788, −2.714341885688298, −1.781430607635352, −1.449857103120128, 0,
1.449857103120128, 1.781430607635352, 2.714341885688298, 3.590404087911788, 3.815738986514928, 4.844200458800341, 5.690550836835603, 5.999580912592519, 6.703235744523883, 7.258165403246340, 8.154376577888664, 8.621460233465600, 8.963891006076287, 9.802754342701026, 10.31542397638784, 10.44757105212995, 11.55450228143933, 11.89143080134049, 12.66321593643214, 13.31171001104949, 13.57949287274309, 14.01429978014087, 14.53598319707156, 15.27122220415730, 15.79143997149466
