| L(s) = 1 | − 2·2-s − 2·3-s + 4-s + 2·5-s + 4·6-s − 9-s − 4·10-s − 2·11-s − 2·12-s − 2·13-s − 4·15-s + 16-s + 2·18-s + 12·19-s + 2·20-s + 4·22-s + 4·23-s − 7·25-s + 4·26-s + 6·27-s − 2·29-s + 8·30-s − 6·31-s + 2·32-s + 4·33-s − 36-s + 8·37-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 1/2·4-s + 0.894·5-s + 1.63·6-s − 1/3·9-s − 1.26·10-s − 0.603·11-s − 0.577·12-s − 0.554·13-s − 1.03·15-s + 1/4·16-s + 0.471·18-s + 2.75·19-s + 0.447·20-s + 0.852·22-s + 0.834·23-s − 7/5·25-s + 0.784·26-s + 1.15·27-s − 0.371·29-s + 1.46·30-s − 1.07·31-s + 0.353·32-s + 0.696·33-s − 1/6·36-s + 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70241161 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70241161 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 17 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 21 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 26 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 109 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 77 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 35 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 - 12 T + 170 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 157 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 122 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 138 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.55555394443246627461767346226, −7.54448852370766495749905109610, −7.01577666339970415855160084319, −6.60777239635069291348114372994, −6.16349688490870618149730380113, −5.88822966665931323324950963302, −5.48682019894797848368521663154, −5.41796346614197815242058929063, −5.03418810724750691092482502469, −4.78119410776502210416417598884, −3.98923856721367180611856356725, −3.66687769318593635213920649347, −3.16190987710716553501087728877, −2.58095099606218524865036942720, −2.51285911260628500985685483077, −1.77530943694735188280772020397, −1.08219177033118217602466893936, −0.984240468573365216731249305720, 0, 0,
0.984240468573365216731249305720, 1.08219177033118217602466893936, 1.77530943694735188280772020397, 2.51285911260628500985685483077, 2.58095099606218524865036942720, 3.16190987710716553501087728877, 3.66687769318593635213920649347, 3.98923856721367180611856356725, 4.78119410776502210416417598884, 5.03418810724750691092482502469, 5.41796346614197815242058929063, 5.48682019894797848368521663154, 5.88822966665931323324950963302, 6.16349688490870618149730380113, 6.60777239635069291348114372994, 7.01577666339970415855160084319, 7.54448852370766495749905109610, 7.55555394443246627461767346226
