| L(s) = 1 | + 24·7-s − 9·9-s + 20·17-s − 96·23-s + 246·25-s + 504·31-s + 188·41-s + 816·47-s − 254·49-s − 216·63-s + 2.11e3·71-s − 660·73-s − 1.22e3·79-s + 81·81-s + 3.02e3·89-s + 1.18e3·97-s + 2.56e3·103-s + 480·119-s − 938·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 180·153-s + 157-s + ⋯ |
| L(s) = 1 | + 1.29·7-s − 1/3·9-s + 0.285·17-s − 0.870·23-s + 1.96·25-s + 2.92·31-s + 0.716·41-s + 2.53·47-s − 0.740·49-s − 0.431·63-s + 3.53·71-s − 1.05·73-s − 1.74·79-s + 1/9·81-s + 3.59·89-s + 1.24·97-s + 2.45·103-s + 0.369·119-s − 0.704·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s − 0.0951·153-s + 0.000508·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.757450484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.757450484\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 246 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 12 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 938 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 2630 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 10 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 3706 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 48 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 2298 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 252 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 29738 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 94 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 107030 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 408 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 178038 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 320758 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 236806 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 559910 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 1056 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 330 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 612 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 825478 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1510 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 594 T + p^{3} T^{2} )^{2} \) |
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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
−10.13750848790233651259637898386, −9.860650393451812073605840082126, −9.133192331489689634739109520712, −8.841644079690861045886325431934, −8.373026623596393182615163496707, −8.056400747990971034441237272552, −7.67630694412453427370816099266, −7.20530376484909630488424465662, −6.54538219189154476288352587696, −6.24872730384168026538577158341, −5.71696070940330646680034780418, −5.09165156661887305566498868084, −4.68657818856085178992026377989, −4.45662264814876764415960950874, −3.67830119488595683645839757111, −3.04082476390225008004202628020, −2.45996763919427474739423338141, −1.96001816914396894937673602014, −0.946628069994315784113882835552, −0.78031552954546111070347212212,
0.78031552954546111070347212212, 0.946628069994315784113882835552, 1.96001816914396894937673602014, 2.45996763919427474739423338141, 3.04082476390225008004202628020, 3.67830119488595683645839757111, 4.45662264814876764415960950874, 4.68657818856085178992026377989, 5.09165156661887305566498868084, 5.71696070940330646680034780418, 6.24872730384168026538577158341, 6.54538219189154476288352587696, 7.20530376484909630488424465662, 7.67630694412453427370816099266, 8.056400747990971034441237272552, 8.373026623596393182615163496707, 8.841644079690861045886325431934, 9.133192331489689634739109520712, 9.860650393451812073605840082126, 10.13750848790233651259637898386
