| L(s) = 1 | + 7·4-s + 30·11-s − 15·16-s − 4·19-s + 60·29-s + 202·31-s + 60·41-s + 210·44-s + 61·49-s − 1.32e3·59-s − 752·61-s − 553·64-s + 720·71-s − 28·76-s − 976·79-s − 900·89-s − 2.85e3·101-s + 1.72e3·109-s + 420·116-s − 1.98e3·121-s + 1.41e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 7/8·4-s + 0.822·11-s − 0.234·16-s − 0.0482·19-s + 0.384·29-s + 1.17·31-s + 0.228·41-s + 0.719·44-s + 0.177·49-s − 2.91·59-s − 1.57·61-s − 1.08·64-s + 1.20·71-s − 0.0422·76-s − 1.38·79-s − 1.07·89-s − 2.80·101-s + 1.51·109-s + 0.336·116-s − 1.49·121-s + 1.02·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.144666941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.144666941\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 7 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 61 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 15 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3994 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4642 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 11338 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 30 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 101 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 83594 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 30 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 146914 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 98746 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 87887 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 660 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 376 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 539026 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 360 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 161809 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 488 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 904453 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 450 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 604321 T^{2} + p^{6} T^{4} \) |
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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
−10.55977588373379161337978235488, −9.776797589004659597619530762294, −9.460089213994318529116190894346, −9.163488454064862400867159730775, −8.473314725623541674266263447268, −8.211171866140290521445409204845, −7.62809615023118115418616423557, −7.16813777591399906857238605129, −6.78414692909597726213672399846, −6.30510148511689376197080258925, −6.03650068644647397142658321893, −5.44674618153580028914462690322, −4.59936366529979248475340369355, −4.46372367720555415729279872542, −3.74941458862351318081897518350, −2.87211501888942111173453804210, −2.82611716667602223455551228512, −1.78688999120370716255889002787, −1.43510304917146763272931857962, −0.49166551717348784367502338878,
0.49166551717348784367502338878, 1.43510304917146763272931857962, 1.78688999120370716255889002787, 2.82611716667602223455551228512, 2.87211501888942111173453804210, 3.74941458862351318081897518350, 4.46372367720555415729279872542, 4.59936366529979248475340369355, 5.44674618153580028914462690322, 6.03650068644647397142658321893, 6.30510148511689376197080258925, 6.78414692909597726213672399846, 7.16813777591399906857238605129, 7.62809615023118115418616423557, 8.211171866140290521445409204845, 8.473314725623541674266263447268, 9.163488454064862400867159730775, 9.460089213994318529116190894346, 9.776797589004659597619530762294, 10.55977588373379161337978235488
