| L(s) = 1 | − 2·2-s + 21·5-s − 8·7-s + 8·8-s − 42·10-s + 36·11-s + 49·13-s + 16·14-s − 16·16-s − 42·17-s − 224·19-s − 72·22-s + 180·23-s + 125·25-s − 98·26-s − 135·29-s − 308·31-s + 84·34-s − 168·35-s − 2·37-s + 448·38-s + 168·40-s − 42·41-s − 20·43-s − 360·46-s + 84·47-s + 343·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.87·5-s − 0.431·7-s + 0.353·8-s − 1.32·10-s + 0.986·11-s + 1.04·13-s + 0.305·14-s − 1/4·16-s − 0.599·17-s − 2.70·19-s − 0.697·22-s + 1.63·23-s + 25-s − 0.739·26-s − 0.864·29-s − 1.78·31-s + 0.423·34-s − 0.811·35-s − 0.00888·37-s + 1.91·38-s + 0.664·40-s − 0.159·41-s − 0.0709·43-s − 1.15·46-s + 0.260·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.967821143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.967821143\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 21 T + 316 T^{2} - 21 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 8 T - 279 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 36 T - 35 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 49 T + 204 T^{2} - 49 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 21 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 112 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 180 T + 20233 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 135 T - 6164 T^{2} + 135 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 19 T + p^{3} T^{2} )( 1 + 289 T + p^{3} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 42 T - 67157 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 20 T - 79107 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 84 T - 96767 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 174 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 504 T + 48637 T^{2} - 504 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 385 T - 78756 T^{2} - 385 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 272 T - 226779 T^{2} + 272 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 888 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 371 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 652 T - 67935 T^{2} - 652 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 84 T - 564731 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 21 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1246 T + 639843 T^{2} - 1246 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−13.19636071814474684010336587790, −12.37365569412686385313093478142, −11.32444565377788754001933627804, −10.83286639728749434244102639692, −10.79647128807278007024727004397, −9.951693221786258885633588973802, −9.575941390545464730437556384030, −8.972271204568524115182737871001, −8.871044944957653067071524676734, −8.344339804060036266096892806714, −7.27378153000945299767131004090, −6.63646743232648757435686047849, −6.41257781912142378158782433183, −5.74627633160831347732966629430, −5.19645933390127768821127762017, −4.13320893001743525914837271162, −3.61725175184557564453483206474, −2.07105184789143414197317796633, −2.03371887894989969365295511358, −0.75160116229901187165317760534,
0.75160116229901187165317760534, 2.03371887894989969365295511358, 2.07105184789143414197317796633, 3.61725175184557564453483206474, 4.13320893001743525914837271162, 5.19645933390127768821127762017, 5.74627633160831347732966629430, 6.41257781912142378158782433183, 6.63646743232648757435686047849, 7.27378153000945299767131004090, 8.344339804060036266096892806714, 8.871044944957653067071524676734, 8.972271204568524115182737871001, 9.575941390545464730437556384030, 9.951693221786258885633588973802, 10.79647128807278007024727004397, 10.83286639728749434244102639692, 11.32444565377788754001933627804, 12.37365569412686385313093478142, 13.19636071814474684010336587790
