L(s) = 1 | − 6·3-s + 16·7-s + 27·9-s + 8·13-s + 12·19-s − 96·21-s + 6·25-s − 108·27-s + 52·31-s + 60·37-s − 48·39-s − 84·43-s + 94·49-s − 72·57-s + 24·61-s + 432·63-s − 4·67-s − 148·73-s − 36·75-s + 80·79-s + 405·81-s + 128·91-s − 312·93-s + 124·97-s − 148·103-s − 400·109-s − 360·111-s + ⋯ |
L(s) = 1 | − 2·3-s + 16/7·7-s + 3·9-s + 8/13·13-s + 0.631·19-s − 4.57·21-s + 6/25·25-s − 4·27-s + 1.67·31-s + 1.62·37-s − 1.23·39-s − 1.95·43-s + 1.91·49-s − 1.26·57-s + 0.393·61-s + 48/7·63-s − 0.0597·67-s − 2.02·73-s − 0.479·75-s + 1.01·79-s + 5·81-s + 1.40·91-s − 3.35·93-s + 1.27·97-s − 1.43·103-s − 3.66·109-s − 3.24·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 278784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 278784 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.868729194\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.868729194\) |
\(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - 6 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 402 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1014 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 98 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3186 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 42 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 3018 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2054 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 2562 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 6518 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 74 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 12194 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 1586 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 62 T + p^{2} T^{2} )^{2} \) |
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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
−10.92620692475161324516011631602, −10.51810821558973494417326180412, −10.33943251028426868990002767479, −9.628932645442603423512893619920, −9.286202169935506125068647791610, −8.429260650455487585054064908035, −7.990771306290242187444338538061, −7.85515545791436087353982109763, −7.16560744760734825664589802660, −6.56829085631142286162290215295, −6.33835001495353085865767011517, −5.55876087665585514855833515599, −5.27938396910351851496202005640, −4.85540258468475393895855049036, −4.36622202926411974006768008516, −4.05142803184477132938254993222, −2.90725838550722139861647276043, −1.75239054810510256747324559388, −1.38267461020345039610994042751, −0.69800252997721015209033229666,
0.69800252997721015209033229666, 1.38267461020345039610994042751, 1.75239054810510256747324559388, 2.90725838550722139861647276043, 4.05142803184477132938254993222, 4.36622202926411974006768008516, 4.85540258468475393895855049036, 5.27938396910351851496202005640, 5.55876087665585514855833515599, 6.33835001495353085865767011517, 6.56829085631142286162290215295, 7.16560744760734825664589802660, 7.85515545791436087353982109763, 7.990771306290242187444338538061, 8.429260650455487585054064908035, 9.286202169935506125068647791610, 9.628932645442603423512893619920, 10.33943251028426868990002767479, 10.51810821558973494417326180412, 10.92620692475161324516011631602