| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s − 2·11-s − 8·13-s + 5·16-s + 6·17-s − 4·22-s − 3·25-s − 16·26-s − 4·29-s − 8·31-s + 6·32-s + 12·34-s − 8·37-s − 18·41-s + 8·43-s − 6·44-s − 8·47-s − 6·50-s − 24·52-s − 8·53-s − 8·58-s + 8·59-s + 8·61-s − 16·62-s + 7·64-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 0.603·11-s − 2.21·13-s + 5/4·16-s + 1.45·17-s − 0.852·22-s − 3/5·25-s − 3.13·26-s − 0.742·29-s − 1.43·31-s + 1.06·32-s + 2.05·34-s − 1.31·37-s − 2.81·41-s + 1.21·43-s − 0.904·44-s − 1.16·47-s − 0.848·50-s − 3.32·52-s − 1.09·53-s − 1.05·58-s + 1.04·59-s + 1.02·61-s − 2.03·62-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 39 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 74 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + p T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 22 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 75 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 178 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 20 T + 218 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16 T + 215 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 10 T + 79 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 16 T + 214 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 83 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−7.29259832148892851819981694914, −7.10526583277237411484371916517, −6.89472688844500263623394192790, −6.57701248164210012835587204339, −5.85317841156047568771201320891, −5.68183884219909896951411680855, −5.28343170654185137238223581323, −5.25832770121441114498776103726, −4.79588018584106298209245175258, −4.53232244814853763024583822035, −3.80923680177797962015715438576, −3.78526277519252135488512400289, −3.18004995943381606656949025677, −3.03379175806038509593832848952, −2.46507451911545987819100337845, −2.07583230112106834888351709169, −1.73059376129159953175002230519, −1.22940663671762592696668762357, 0, 0,
1.22940663671762592696668762357, 1.73059376129159953175002230519, 2.07583230112106834888351709169, 2.46507451911545987819100337845, 3.03379175806038509593832848952, 3.18004995943381606656949025677, 3.78526277519252135488512400289, 3.80923680177797962015715438576, 4.53232244814853763024583822035, 4.79588018584106298209245175258, 5.25832770121441114498776103726, 5.28343170654185137238223581323, 5.68183884219909896951411680855, 5.85317841156047568771201320891, 6.57701248164210012835587204339, 6.89472688844500263623394192790, 7.10526583277237411484371916517, 7.29259832148892851819981694914
