| L(s) = 1 | − 2-s − 2·4-s + 2·7-s + 3·8-s − 4·13-s − 2·14-s + 16-s − 4·17-s + 6·19-s + 4·23-s − 5·25-s + 4·26-s − 4·28-s − 10·29-s + 12·31-s − 2·32-s + 4·34-s + 6·37-s − 6·38-s + 4·41-s − 8·43-s − 4·46-s − 10·47-s + 3·49-s + 5·50-s + 8·52-s − 4·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 4-s + 0.755·7-s + 1.06·8-s − 1.10·13-s − 0.534·14-s + 1/4·16-s − 0.970·17-s + 1.37·19-s + 0.834·23-s − 25-s + 0.784·26-s − 0.755·28-s − 1.85·29-s + 2.15·31-s − 0.353·32-s + 0.685·34-s + 0.986·37-s − 0.973·38-s + 0.624·41-s − 1.21·43-s − 0.589·46-s − 1.45·47-s + 3/7·49-s + 0.707·50-s + 1.10·52-s − 0.549·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 27 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 63 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 10 T + 99 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 39 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 - 16 T + 166 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 117 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 154 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 24 T + 290 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 174 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.83010232123211206288182724240, −7.49998766188780965785890376124, −7.09504191888256671674590640983, −6.79197779603329972875071808828, −6.45263885359085466676699885698, −5.85069627749114426552617409701, −5.39375707161474817378761251087, −5.32656563552552108396634140195, −4.83081410640440801215156267248, −4.55171299638589287352044279562, −4.12486850470002047361570542622, −3.99050152611433470843246374556, −3.26868848174133350674425819847, −2.79113172341843901723743976346, −2.51662328984033359975041980738, −1.89498610757765272751121870617, −1.30252169858491426478512536069, −1.04740794424560128598230650101, 0, 0,
1.04740794424560128598230650101, 1.30252169858491426478512536069, 1.89498610757765272751121870617, 2.51662328984033359975041980738, 2.79113172341843901723743976346, 3.26868848174133350674425819847, 3.99050152611433470843246374556, 4.12486850470002047361570542622, 4.55171299638589287352044279562, 4.83081410640440801215156267248, 5.32656563552552108396634140195, 5.39375707161474817378761251087, 5.85069627749114426552617409701, 6.45263885359085466676699885698, 6.79197779603329972875071808828, 7.09504191888256671674590640983, 7.49998766188780965785890376124, 7.83010232123211206288182724240
