L(s) = 1 | − 2·5-s + 2·7-s + 4·11-s + 6·17-s + 8·19-s − 8·23-s + 25-s − 8·31-s − 4·35-s − 6·37-s + 2·41-s + 10·43-s + 6·47-s + 3·49-s − 4·53-s − 8·55-s − 10·59-s + 12·61-s + 8·67-s − 8·71-s + 8·77-s + 2·79-s − 22·83-s − 12·85-s − 12·89-s − 16·95-s − 8·97-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 0.755·7-s + 1.20·11-s + 1.45·17-s + 1.83·19-s − 1.66·23-s + 1/5·25-s − 1.43·31-s − 0.676·35-s − 0.986·37-s + 0.312·41-s + 1.52·43-s + 0.875·47-s + 3/7·49-s − 0.549·53-s − 1.07·55-s − 1.30·59-s + 1.53·61-s + 0.977·67-s − 0.949·71-s + 0.911·77-s + 0.225·79-s − 2.41·83-s − 1.30·85-s − 1.27·89-s − 1.64·95-s − 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36578304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36578304 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.986964266\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.986964266\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 5 | $D_{4}$ | \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 51 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 75 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 103 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 71 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 111 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 151 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 22 T + 255 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 10 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.082296960076888904802630217763, −8.005536989658596281768854452131, −7.44551855835553024727235142303, −7.32243724935105443483460157591, −7.01317174147959844978427132310, −6.62188133774341958487151971261, −5.81874442639145192198258877998, −5.76350983504975032690343745459, −5.54269197240594627474143799295, −5.09120699371429884834835434206, −4.40720694398610572554533319256, −4.26252042643615297038216576122, −3.79283261301605203107269775773, −3.60431337725837845947277598668, −3.05786400232492525043737300232, −2.71897961403163818982807059730, −1.78683978135011886474510670710, −1.68318988369617434884633142313, −1.03821271148208423175406920275, −0.50791667873882477997765791007,
0.50791667873882477997765791007, 1.03821271148208423175406920275, 1.68318988369617434884633142313, 1.78683978135011886474510670710, 2.71897961403163818982807059730, 3.05786400232492525043737300232, 3.60431337725837845947277598668, 3.79283261301605203107269775773, 4.26252042643615297038216576122, 4.40720694398610572554533319256, 5.09120699371429884834835434206, 5.54269197240594627474143799295, 5.76350983504975032690343745459, 5.81874442639145192198258877998, 6.62188133774341958487151971261, 7.01317174147959844978427132310, 7.32243724935105443483460157591, 7.44551855835553024727235142303, 8.005536989658596281768854452131, 8.082296960076888904802630217763