| L(s) = 1 | + 2-s − 2·4-s + 2·5-s − 2·7-s − 3·8-s + 2·10-s − 4·13-s − 2·14-s + 16-s − 10·19-s − 4·20-s + 2·23-s − 2·25-s − 4·26-s + 4·28-s − 6·29-s − 18·31-s + 2·32-s − 4·35-s + 2·37-s − 10·38-s − 6·40-s − 4·43-s + 2·46-s − 2·47-s + 3·49-s − 2·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 4-s + 0.894·5-s − 0.755·7-s − 1.06·8-s + 0.632·10-s − 1.10·13-s − 0.534·14-s + 1/4·16-s − 2.29·19-s − 0.894·20-s + 0.417·23-s − 2/5·25-s − 0.784·26-s + 0.755·28-s − 1.11·29-s − 3.23·31-s + 0.353·32-s − 0.676·35-s + 0.328·37-s − 1.62·38-s − 0.948·40-s − 0.609·43-s + 0.294·46-s − 0.291·47-s + 3/7·49-s − 0.282·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2099601 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2099601 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 77 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 18 T + 182 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 201 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 101 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 178 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 158 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−9.385085086192917205394596770999, −9.127546393489788368531209178576, −8.579570551931178091073449839404, −8.275339229130114844571621749071, −7.65266912194693059905827471533, −7.25169311646161658193193904206, −6.63515169794237612026554356861, −6.54278718488329100826611977558, −5.68991149007027034279761966609, −5.67685022159794092375917813692, −5.14278704436130463271167253404, −4.77023492345704792891189718122, −4.04868621031737396334402956741, −3.94891665698591543283003876991, −3.38299350437100370859341759677, −2.68912544268589076509148654148, −2.05968873342057793782048375427, −1.70468650874728080258833059320, 0, 0,
1.70468650874728080258833059320, 2.05968873342057793782048375427, 2.68912544268589076509148654148, 3.38299350437100370859341759677, 3.94891665698591543283003876991, 4.04868621031737396334402956741, 4.77023492345704792891189718122, 5.14278704436130463271167253404, 5.67685022159794092375917813692, 5.68991149007027034279761966609, 6.54278718488329100826611977558, 6.63515169794237612026554356861, 7.25169311646161658193193904206, 7.65266912194693059905827471533, 8.275339229130114844571621749071, 8.579570551931178091073449839404, 9.127546393489788368531209178576, 9.385085086192917205394596770999
