| L(s) = 1 | − 2·7-s − 4·11-s + 2·13-s − 2·19-s + 4·23-s − 4·29-s + 12·31-s − 10·37-s − 4·41-s + 12·43-s − 11·49-s − 4·53-s − 2·61-s + 10·67-s − 28·71-s − 2·73-s + 8·77-s + 2·79-s + 20·83-s − 24·89-s − 4·91-s − 22·97-s − 24·101-s − 2·103-s − 4·107-s − 20·113-s − 4·121-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1.20·11-s + 0.554·13-s − 0.458·19-s + 0.834·23-s − 0.742·29-s + 2.15·31-s − 1.64·37-s − 0.624·41-s + 1.82·43-s − 1.57·49-s − 0.549·53-s − 0.256·61-s + 1.22·67-s − 3.32·71-s − 0.234·73-s + 0.911·77-s + 0.225·79-s + 2.19·83-s − 2.54·89-s − 0.419·91-s − 2.23·97-s − 2.38·101-s − 0.197·103-s − 0.386·107-s − 1.88·113-s − 0.363·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 44 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 56 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 10 T + 75 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 32 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 104 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 10 T + 63 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 28 T + 332 T^{2} + 28 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 123 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 135 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 20 T + 260 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 22 T + 3 p T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
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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
−7.963459057594579122403119533656, −7.78829570250052895617560346284, −7.17945594294472859161581297922, −6.90653048493958958531445178052, −6.60335726409188197834078815252, −6.26426364323389455850012260092, −5.76295786303059734440524597597, −5.58229840793489948713227592113, −4.92810387794930377824471525324, −4.92577956599006983336175910839, −4.13426275597960821572306435630, −4.06068231896071888977684577272, −3.23756246352056233385789328412, −3.18094941228393760994396172663, −2.48212152249939269578907577761, −2.46396904581188590258298956592, −1.32513231348801711207744572949, −1.32399054697859197599549013288, 0, 0,
1.32399054697859197599549013288, 1.32513231348801711207744572949, 2.46396904581188590258298956592, 2.48212152249939269578907577761, 3.18094941228393760994396172663, 3.23756246352056233385789328412, 4.06068231896071888977684577272, 4.13426275597960821572306435630, 4.92577956599006983336175910839, 4.92810387794930377824471525324, 5.58229840793489948713227592113, 5.76295786303059734440524597597, 6.26426364323389455850012260092, 6.60335726409188197834078815252, 6.90653048493958958531445178052, 7.17945594294472859161581297922, 7.78829570250052895617560346284, 7.963459057594579122403119533656
