L(s) = 1 | − 3-s − 5-s + 9-s + 4·11-s + 2·13-s + 15-s − 2·17-s + 4·19-s + 25-s − 27-s − 2·29-s − 4·33-s − 10·37-s − 2·39-s − 10·41-s − 4·43-s − 45-s + 8·47-s + 2·51-s − 10·53-s − 4·55-s − 4·57-s − 4·59-s + 2·61-s − 2·65-s − 12·67-s + 8·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.258·15-s − 0.485·17-s + 0.917·19-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.696·33-s − 1.64·37-s − 0.320·39-s − 1.56·41-s − 0.609·43-s − 0.149·45-s + 1.16·47-s + 0.280·51-s − 1.37·53-s − 0.539·55-s − 0.529·57-s − 0.520·59-s + 0.256·61-s − 0.248·65-s − 1.46·67-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−16.69136433840718, −16.16990187523591, −15.53347536741930, −15.18881291390267, −14.38345157409689, −13.80967576615606, −13.35728670838382, −12.51948401880638, −11.91728655874259, −11.70087374583685, −10.96482367480647, −10.47261468383090, −9.701987193460744, −9.050074368120516, −8.585660223233458, −7.751701395713314, −7.045611696379868, −6.592632873218824, −5.908844180382647, −5.145878649181373, −4.499031012015956, −3.688528310417987, −3.230193859038328, −1.864671260896310, −1.170588651737777, 0,
1.170588651737777, 1.864671260896310, 3.230193859038328, 3.688528310417987, 4.499031012015956, 5.145878649181373, 5.908844180382647, 6.592632873218824, 7.045611696379868, 7.751701395713314, 8.585660223233458, 9.050074368120516, 9.701987193460744, 10.47261468383090, 10.96482367480647, 11.70087374583685, 11.91728655874259, 12.51948401880638, 13.35728670838382, 13.80967576615606, 14.38345157409689, 15.18881291390267, 15.53347536741930, 16.16990187523591, 16.69136433840718