| L(s) = 1 | − 7-s + 2·17-s + 4·19-s − 4·23-s − 5·25-s − 6·29-s + 4·31-s + 6·37-s + 6·41-s + 4·43-s − 12·47-s + 49-s + 4·53-s − 8·61-s + 12·67-s + 4·71-s + 4·73-s − 4·83-s + 12·89-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 2·119-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.485·17-s + 0.917·19-s − 0.834·23-s − 25-s − 1.11·29-s + 0.718·31-s + 0.986·37-s + 0.937·41-s + 0.609·43-s − 1.75·47-s + 1/7·49-s + 0.549·53-s − 1.02·61-s + 1.46·67-s + 0.474·71-s + 0.468·73-s − 0.439·83-s + 1.27·89-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.183·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.49284823033570, −14.03305741890789, −13.56733217936818, −13.00401712343349, −12.61519745347785, −11.87378124168215, −11.65662981173203, −11.03545431712424, −10.43670996297599, −9.770207101438334, −9.564737204681689, −9.058608436136889, −8.144161110681061, −7.815434807616716, −7.430509914530488, −6.508672479927287, −6.241136803571127, −5.466538207895440, −5.158964966825104, −4.117844702630346, −3.885090305505909, −3.070802653929850, −2.489317902165030, −1.705566109190350, −0.9203624637792201, 0,
0.9203624637792201, 1.705566109190350, 2.489317902165030, 3.070802653929850, 3.885090305505909, 4.117844702630346, 5.158964966825104, 5.466538207895440, 6.241136803571127, 6.508672479927287, 7.430509914530488, 7.815434807616716, 8.144161110681061, 9.058608436136889, 9.564737204681689, 9.770207101438334, 10.43670996297599, 11.03545431712424, 11.65662981173203, 11.87378124168215, 12.61519745347785, 13.00401712343349, 13.56733217936818, 14.03305741890789, 14.49284823033570
