| L(s) = 1 | − 4·4-s + 50·9-s + 96·11-s + 16·16-s − 4·19-s + 108·29-s + 472·31-s − 200·36-s + 252·41-s − 384·44-s − 49·49-s − 276·59-s + 760·61-s − 64·64-s + 1.15e3·71-s + 16·76-s − 1.55e3·79-s + 1.77e3·81-s + 780·89-s + 4.80e3·99-s − 3.00e3·101-s − 292·109-s − 432·116-s + 4.25e3·121-s − 1.88e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.85·9-s + 2.63·11-s + 1/4·16-s − 0.0482·19-s + 0.691·29-s + 2.73·31-s − 0.925·36-s + 0.959·41-s − 1.31·44-s − 1/7·49-s − 0.609·59-s + 1.59·61-s − 1/8·64-s + 1.92·71-s + 0.0241·76-s − 2.21·79-s + 2.42·81-s + 0.928·89-s + 4.87·99-s − 2.95·101-s − 0.256·109-s − 0.345·116-s + 3.19·121-s − 1.36·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.245576352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.245576352\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 50 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 48 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 1258 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3170 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 9934 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 54 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 236 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 79990 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 126 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 17638 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 207502 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 267478 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 138 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 380 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 367270 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 576 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 544466 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 776 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1000690 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 390 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 56446 T^{2} + p^{6} T^{4} \) |
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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
−11.27176387325703175480480359388, −10.87857352846391067909602504186, −10.10312861385055011517195769606, −9.871861964623956767925877347370, −9.516133513723247689336917317952, −9.141892872587347474965372533464, −8.370075624885830623104794391735, −8.253569738584311165547224596943, −7.35279994130723980741640766461, −6.93457278721706415836028475405, −6.46706043510843277391862424417, −6.25619259843792671186564031360, −5.30888770507981326710275905244, −4.41119692384933414134900245077, −4.38002565143775987686512572853, −3.89425978743647049505886790607, −3.11104108105476143660962820973, −2.05377431655629777348620308821, −1.06566724600104882242188777918, −1.03687983351471668440796864194,
1.03687983351471668440796864194, 1.06566724600104882242188777918, 2.05377431655629777348620308821, 3.11104108105476143660962820973, 3.89425978743647049505886790607, 4.38002565143775987686512572853, 4.41119692384933414134900245077, 5.30888770507981326710275905244, 6.25619259843792671186564031360, 6.46706043510843277391862424417, 6.93457278721706415836028475405, 7.35279994130723980741640766461, 8.253569738584311165547224596943, 8.370075624885830623104794391735, 9.141892872587347474965372533464, 9.516133513723247689336917317952, 9.871861964623956767925877347370, 10.10312861385055011517195769606, 10.87857352846391067909602504186, 11.27176387325703175480480359388
