| L(s) = 1 | − 2·4-s + 2·7-s − 4·13-s + 6·17-s − 8·19-s − 8·25-s − 4·28-s + 6·29-s + 8·31-s + 6·37-s + 6·41-s − 20·43-s + 12·47-s + 3·49-s + 8·52-s + 18·53-s − 16·61-s + 8·64-s − 12·68-s − 12·71-s + 4·73-s + 16·76-s − 22·79-s − 12·83-s − 24·89-s − 8·91-s + 14·97-s + ⋯ |
| L(s) = 1 | − 4-s + 0.755·7-s − 1.10·13-s + 1.45·17-s − 1.83·19-s − 8/5·25-s − 0.755·28-s + 1.11·29-s + 1.43·31-s + 0.986·37-s + 0.937·41-s − 3.04·43-s + 1.75·47-s + 3/7·49-s + 1.10·52-s + 2.47·53-s − 2.04·61-s + 64-s − 1.45·68-s − 1.42·71-s + 0.468·73-s + 1.83·76-s − 2.47·79-s − 1.31·83-s − 2.54·89-s − 0.838·91-s + 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64016001 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64016001 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 127 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 36 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 65 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 18 T + 185 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 16 T + 168 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 176 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T - 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 170 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 24 T + 314 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 225 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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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
−7.75320477146777425783092406739, −7.31906698804757137037138337993, −7.05513764302992564170698937546, −6.72934086995647725577969969948, −6.03684823724673602924439763536, −5.98167983232346155510781019310, −5.55359961472051183615428166020, −5.19771256057697818301319066712, −4.60708170040044033590254896217, −4.59550213354403167723453818739, −4.11659623374410123887875248495, −4.02280591851302719333729214103, −3.33643374440277882688963990821, −2.72467201976063966798834257504, −2.53874983184263369065480308388, −2.09158806736332006898035817641, −1.30485852493294811096156507333, −1.13271918651474050063146331522, 0, 0,
1.13271918651474050063146331522, 1.30485852493294811096156507333, 2.09158806736332006898035817641, 2.53874983184263369065480308388, 2.72467201976063966798834257504, 3.33643374440277882688963990821, 4.02280591851302719333729214103, 4.11659623374410123887875248495, 4.59550213354403167723453818739, 4.60708170040044033590254896217, 5.19771256057697818301319066712, 5.55359961472051183615428166020, 5.98167983232346155510781019310, 6.03684823724673602924439763536, 6.72934086995647725577969969948, 7.05513764302992564170698937546, 7.31906698804757137037138337993, 7.75320477146777425783092406739
