| L(s) = 1 | − 2·4-s − 2·7-s + 34·13-s + 4·16-s − 62·19-s + 4·28-s − 62·31-s − 32·37-s + 130·43-s − 95·49-s − 68·52-s − 110·61-s − 8·64-s − 146·67-s + 112·73-s + 124·76-s + 208·79-s − 68·91-s + 178·97-s − 164·103-s + 142·109-s − 8·112-s + 224·121-s + 124·124-s + 127-s + 131-s + 124·133-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 2/7·7-s + 2.61·13-s + 1/4·16-s − 3.26·19-s + 1/7·28-s − 2·31-s − 0.864·37-s + 3.02·43-s − 1.93·49-s − 1.30·52-s − 1.80·61-s − 1/8·64-s − 2.17·67-s + 1.53·73-s + 1.63·76-s + 2.63·79-s − 0.747·91-s + 1.83·97-s − 1.59·103-s + 1.30·109-s − 0.0714·112-s + 1.85·121-s + 124-s + 0.00787·127-s + 0.00763·131-s + 0.932·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.389944173\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.389944173\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 224 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 17 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 304 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 31 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1040 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 224 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3074 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 65 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3968 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 5330 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 6160 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 55 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 73 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 4030 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 56 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 104 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 13616 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 14690 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 89 T + p^{2} T^{2} )^{2} \) |
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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
−10.88841961777572520168006882496, −10.78278328174507392435443954747, −10.52058960513546600597722482348, −9.623703901323081708092429389041, −9.105631289517625272778830482989, −8.876064637306440186455517829815, −8.554517719041407922348680706754, −8.002744656616821525125191752352, −7.58076056947763486615636859543, −6.70309396353947817733083776783, −6.42020472157017055168240855793, −5.88372829689165320131782795935, −5.70080750738742012677563773114, −4.46840374650707330201169201161, −4.45265821680635544410746634243, −3.55891055156435265022854934300, −3.42591282247986109923825212205, −2.16285690686954947771582296698, −1.63913962567493917656566909809, −0.49965708849887783610113584457,
0.49965708849887783610113584457, 1.63913962567493917656566909809, 2.16285690686954947771582296698, 3.42591282247986109923825212205, 3.55891055156435265022854934300, 4.45265821680635544410746634243, 4.46840374650707330201169201161, 5.70080750738742012677563773114, 5.88372829689165320131782795935, 6.42020472157017055168240855793, 6.70309396353947817733083776783, 7.58076056947763486615636859543, 8.002744656616821525125191752352, 8.554517719041407922348680706754, 8.876064637306440186455517829815, 9.105631289517625272778830482989, 9.623703901323081708092429389041, 10.52058960513546600597722482348, 10.78278328174507392435443954747, 10.88841961777572520168006882496
