L(s) = 1 | − 2·3-s − 2·5-s − 10·13-s + 4·15-s + 2·17-s + 6·23-s + 3·25-s + 2·27-s + 4·29-s − 4·31-s + 10·37-s + 20·39-s − 4·41-s − 4·43-s − 4·47-s − 2·49-s − 4·51-s + 14·53-s − 16·59-s + 4·61-s + 20·65-s − 6·67-s − 12·69-s + 12·71-s + 12·73-s − 6·75-s − 81-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.894·5-s − 2.77·13-s + 1.03·15-s + 0.485·17-s + 1.25·23-s + 3/5·25-s + 0.384·27-s + 0.742·29-s − 0.718·31-s + 1.64·37-s + 3.20·39-s − 0.624·41-s − 0.609·43-s − 0.583·47-s − 2/7·49-s − 0.560·51-s + 1.92·53-s − 2.08·59-s + 0.512·61-s + 2.48·65-s − 0.733·67-s − 1.44·69-s + 1.42·71-s + 1.40·73-s − 0.692·75-s − 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64320400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64320400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9215085641\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9215085641\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 401 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 10 T + 48 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 32 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 50 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 10 T + 72 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T - 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 50 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 14 T + 128 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 116 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 4 T - 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 18 T + 272 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
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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
−7.74733845697301178225989411630, −7.63079997405439544254797238383, −7.29684120061152659578309552910, −7.06325261747275529071148803104, −6.59851075117276055085247849550, −6.39004813766133549467414595153, −5.72236520779629232424433432274, −5.64971065936200033759038193575, −5.14133464191926448945115820768, −4.81313050260009395957652222512, −4.55854777703623690564683632452, −4.49391564219667765674273899465, −3.48958739474804248679747300250, −3.44482908635180680346859506415, −2.94462329159905247246810262698, −2.33157528550292942006398605068, −2.22968582082182138094575734697, −1.35341986370336354540169973606, −0.57271194669710679835503882252, −0.44951363442382159503491089694,
0.44951363442382159503491089694, 0.57271194669710679835503882252, 1.35341986370336354540169973606, 2.22968582082182138094575734697, 2.33157528550292942006398605068, 2.94462329159905247246810262698, 3.44482908635180680346859506415, 3.48958739474804248679747300250, 4.49391564219667765674273899465, 4.55854777703623690564683632452, 4.81313050260009395957652222512, 5.14133464191926448945115820768, 5.64971065936200033759038193575, 5.72236520779629232424433432274, 6.39004813766133549467414595153, 6.59851075117276055085247849550, 7.06325261747275529071148803104, 7.29684120061152659578309552910, 7.63079997405439544254797238383, 7.74733845697301178225989411630