L(s) = 1 | − 2·5-s + 6·7-s + 2·13-s − 2·19-s − 4·23-s − 2·25-s − 8·29-s + 14·31-s − 12·35-s + 14·41-s − 4·43-s − 8·47-s + 18·49-s − 8·53-s + 4·59-s − 16·61-s − 4·65-s + 2·67-s − 20·71-s + 12·83-s + 18·89-s + 12·91-s + 4·95-s + 24·97-s + 20·101-s + 4·103-s + 4·107-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 2.26·7-s + 0.554·13-s − 0.458·19-s − 0.834·23-s − 2/5·25-s − 1.48·29-s + 2.51·31-s − 2.02·35-s + 2.18·41-s − 0.609·43-s − 1.16·47-s + 18/7·49-s − 1.09·53-s + 0.520·59-s − 2.04·61-s − 0.496·65-s + 0.244·67-s − 2.37·71-s + 1.31·83-s + 1.90·89-s + 1.25·91-s + 0.410·95-s + 2.43·97-s + 1.99·101-s + 0.394·103-s + 0.386·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56070144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56070144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.865296771\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.865296771\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 - 14 T + 106 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 14 T + 126 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 90 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 214 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 24 T + 318 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
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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
−7.926489408460554011770534644484, −7.85459387459277641648216490126, −7.54049714042846935094025442442, −7.33080727485402191519169657598, −6.51714408628138160579642281118, −6.36966337736594870299495148920, −5.87780850815003785958240140050, −5.79470334361140405204788218373, −4.93270581481378579724642753458, −4.90870218899717124800131522706, −4.44867801724661209654420009572, −4.35857028531152284536671161288, −3.73350579332389387793333539906, −3.52965832653256396852235581183, −2.88802539733523814497018709873, −2.41966596860460804452903040820, −1.77520191729155550262632348087, −1.73299118745867855814196444418, −1.04057333024007474020774710321, −0.44470686000457059600434419072,
0.44470686000457059600434419072, 1.04057333024007474020774710321, 1.73299118745867855814196444418, 1.77520191729155550262632348087, 2.41966596860460804452903040820, 2.88802539733523814497018709873, 3.52965832653256396852235581183, 3.73350579332389387793333539906, 4.35857028531152284536671161288, 4.44867801724661209654420009572, 4.90870218899717124800131522706, 4.93270581481378579724642753458, 5.79470334361140405204788218373, 5.87780850815003785958240140050, 6.36966337736594870299495148920, 6.51714408628138160579642281118, 7.33080727485402191519169657598, 7.54049714042846935094025442442, 7.85459387459277641648216490126, 7.926489408460554011770534644484