| L(s) = 1 | + 3·4-s − 2·7-s − 4·13-s + 5·16-s + 14·19-s − 3·25-s − 6·28-s + 6·31-s − 6·37-s + 16·43-s + 3·49-s − 12·52-s − 16·61-s + 3·64-s − 4·67-s + 42·76-s − 8·79-s + 8·91-s − 24·97-s − 9·100-s − 26·103-s + 18·109-s − 10·112-s − 15·121-s + 18·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 0.755·7-s − 1.10·13-s + 5/4·16-s + 3.21·19-s − 3/5·25-s − 1.13·28-s + 1.07·31-s − 0.986·37-s + 2.43·43-s + 3/7·49-s − 1.66·52-s − 2.04·61-s + 3/8·64-s − 0.488·67-s + 4.81·76-s − 0.900·79-s + 0.838·91-s − 2.43·97-s − 0.899·100-s − 2.56·103-s + 1.72·109-s − 0.944·112-s − 1.36·121-s + 1.61·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.696124929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.696124929\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 75 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 79 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 165 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
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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
−12.35921712284478852160771041755, −12.19387318073910906895310642034, −12.09232353811379904909837338543, −11.22973633454396579993969170855, −11.11892413499251378716099938820, −10.19859789402338506821667194321, −9.993472224404305896523652403890, −9.427558516375359699621925509862, −9.025486334051499108478195581285, −7.942263951458807091581982197439, −7.52651246855958083034618111272, −7.24597799911050695308721606126, −6.71919753984202071510762151020, −5.95373906459158470566725457933, −5.58688638814471852323610914395, −4.84768384315861066489256786100, −3.81585215582383374941550744524, −2.86589602692304008188598883868, −2.74106997503221499804348918654, −1.37701167359795407767938570487,
1.37701167359795407767938570487, 2.74106997503221499804348918654, 2.86589602692304008188598883868, 3.81585215582383374941550744524, 4.84768384315861066489256786100, 5.58688638814471852323610914395, 5.95373906459158470566725457933, 6.71919753984202071510762151020, 7.24597799911050695308721606126, 7.52651246855958083034618111272, 7.942263951458807091581982197439, 9.025486334051499108478195581285, 9.427558516375359699621925509862, 9.993472224404305896523652403890, 10.19859789402338506821667194321, 11.11892413499251378716099938820, 11.22973633454396579993969170855, 12.09232353811379904909837338543, 12.19387318073910906895310642034, 12.35921712284478852160771041755
