| L(s) = 1 | + 38·9-s + 72·11-s + 232·19-s − 396·29-s + 480·31-s + 884·41-s + 430·49-s + 696·59-s − 1.14e3·61-s + 336·71-s − 1.56e3·79-s + 715·81-s − 2.06e3·89-s + 2.73e3·99-s − 1.34e3·101-s − 2.09e3·109-s + 1.22e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.63e3·169-s + ⋯ |
| L(s) = 1 | + 1.40·9-s + 1.97·11-s + 2.80·19-s − 2.53·29-s + 2.78·31-s + 3.36·41-s + 1.25·49-s + 1.53·59-s − 2.39·61-s + 0.561·71-s − 2.23·79-s + 0.980·81-s − 2.46·89-s + 2.77·99-s − 1.32·101-s − 1.83·109-s + 0.921·121-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 + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.19·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.805811232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.805811232\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 38 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 430 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 36 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 2630 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2274 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 116 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 24078 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 198 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 240 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 34742 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 442 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 73750 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 53982 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 277590 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 348 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 570 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 122662 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 168 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 760078 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 784 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 825478 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1034 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1679422 T^{2} + p^{6} T^{4} \) |
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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
−12.02343260384702402278344696717, −11.86973079360534637925841458787, −11.42208020172788425029033100771, −10.85500463330507330897944622810, −10.15745899440670877156706598868, −9.603590896005451070985108654638, −9.344092769090138442872067354975, −9.145221925942046495267621399024, −8.040286713541801105926705150754, −7.65983020098503821694625706229, −6.98824900615930169016160018816, −6.88165250934818917837673443514, −5.76999738444481152293212027611, −5.66419369027555156245143587564, −4.34862853515846421998254977955, −4.28802926869621731914703739714, −3.49350249336524233162422394896, −2.61048317056975032243963418305, −1.21199144134572673662735758464, −1.16970272847247722146240742882,
1.16970272847247722146240742882, 1.21199144134572673662735758464, 2.61048317056975032243963418305, 3.49350249336524233162422394896, 4.28802926869621731914703739714, 4.34862853515846421998254977955, 5.66419369027555156245143587564, 5.76999738444481152293212027611, 6.88165250934818917837673443514, 6.98824900615930169016160018816, 7.65983020098503821694625706229, 8.040286713541801105926705150754, 9.145221925942046495267621399024, 9.344092769090138442872067354975, 9.603590896005451070985108654638, 10.15745899440670877156706598868, 10.85500463330507330897944622810, 11.42208020172788425029033100771, 11.86973079360534637925841458787, 12.02343260384702402278344696717
