| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s + 2·5-s − 4·6-s + 4·8-s + 3·9-s + 4·10-s − 6·11-s − 6·12-s − 2·13-s − 4·15-s + 5·16-s + 2·17-s + 6·18-s + 4·19-s + 6·20-s − 12·22-s − 12·23-s − 8·24-s − 7·25-s − 4·26-s − 4·27-s − 8·29-s − 8·30-s + 8·31-s + 6·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s + 1.41·8-s + 9-s + 1.26·10-s − 1.80·11-s − 1.73·12-s − 0.554·13-s − 1.03·15-s + 5/4·16-s + 0.485·17-s + 1.41·18-s + 0.917·19-s + 1.34·20-s − 2.55·22-s − 2.50·23-s − 1.63·24-s − 7/5·25-s − 0.784·26-s − 0.769·27-s − 1.48·29-s − 1.46·30-s + 1.43·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24980004 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24980004 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 29 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 + 12 T + 74 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 72 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 60 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 10 T + 67 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 68 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 109 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 16 T + 140 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 99 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 61 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T - 4 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 14 T + 163 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 181 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 14 T + 213 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 22 T + 3 p T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 195 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−7.900265414227790271720039308260, −7.60320776955292366652285919985, −7.22104362480009141757129153837, −6.86230406707792879823196882531, −6.34056837063142664947094121074, −6.15324973301255640771309066402, −5.63737658810082109413467162261, −5.55216263015654737539146589480, −5.20822995222430638991053522901, −5.02097481754984035721534323951, −4.42127110474716407397690003477, −4.07195053133499393012173342885, −3.58707330245954172800683386957, −3.21935439044866213925113963056, −2.54765220617574294118133168855, −2.29603981038630018112668121140, −1.62554399514333049626924143747, −1.51944590399535319527432851602, 0, 0,
1.51944590399535319527432851602, 1.62554399514333049626924143747, 2.29603981038630018112668121140, 2.54765220617574294118133168855, 3.21935439044866213925113963056, 3.58707330245954172800683386957, 4.07195053133499393012173342885, 4.42127110474716407397690003477, 5.02097481754984035721534323951, 5.20822995222430638991053522901, 5.55216263015654737539146589480, 5.63737658810082109413467162261, 6.15324973301255640771309066402, 6.34056837063142664947094121074, 6.86230406707792879823196882531, 7.22104362480009141757129153837, 7.60320776955292366652285919985, 7.900265414227790271720039308260
