| L(s) = 1 | − 2-s + 4·5-s + 8-s − 4·10-s − 4·11-s + 8·13-s − 16-s − 4·19-s + 4·22-s + 5·25-s − 8·26-s − 4·29-s − 8·31-s + 6·37-s + 4·38-s + 4·40-s + 8·43-s − 8·47-s − 5·50-s − 10·53-s − 16·55-s + 4·58-s + 4·59-s + 4·61-s + 8·62-s + 64-s + 32·65-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.78·5-s + 0.353·8-s − 1.26·10-s − 1.20·11-s + 2.21·13-s − 1/4·16-s − 0.917·19-s + 0.852·22-s + 25-s − 1.56·26-s − 0.742·29-s − 1.43·31-s + 0.986·37-s + 0.648·38-s + 0.632·40-s + 1.21·43-s − 1.16·47-s − 0.707·50-s − 1.37·53-s − 2.15·55-s + 0.525·58-s + 0.520·59-s + 0.512·61-s + 1.01·62-s + 1/8·64-s + 3.96·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.764133591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.764133591\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 10 T + 47 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 16 T + 183 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 8 T - 25 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−10.48695364713228153982365501072, −9.685257599676271051346700399840, −9.652807071533585852443385720377, −9.016897024190777602773890202481, −8.869751579219201565531235440660, −8.381414782025064023091538801546, −7.74374233983636312204040023339, −7.68549050853452138307456568301, −6.89611641967353432819546507361, −6.14977336064038465744980275039, −6.07707403023140270209581531075, −5.88905667869215440445293324459, −4.95043642347979505470465316784, −4.91894488975789101814639869503, −3.79401140698275388105745051530, −3.58882158816331300256097338258, −2.66809605167223772175149865591, −1.95031385945644769042964023597, −1.76868577489678485872342207584, −0.72826011317987794711257719664,
0.72826011317987794711257719664, 1.76868577489678485872342207584, 1.95031385945644769042964023597, 2.66809605167223772175149865591, 3.58882158816331300256097338258, 3.79401140698275388105745051530, 4.91894488975789101814639869503, 4.95043642347979505470465316784, 5.88905667869215440445293324459, 6.07707403023140270209581531075, 6.14977336064038465744980275039, 6.89611641967353432819546507361, 7.68549050853452138307456568301, 7.74374233983636312204040023339, 8.381414782025064023091538801546, 8.869751579219201565531235440660, 9.016897024190777602773890202481, 9.652807071533585852443385720377, 9.685257599676271051346700399840, 10.48695364713228153982365501072
