| L(s) = 1 | − 2·2-s + 3·4-s + 4·5-s − 3·7-s − 4·8-s − 3·9-s − 8·10-s − 5·11-s − 12·13-s + 6·14-s + 5·16-s − 3·17-s + 6·18-s − 5·19-s + 12·20-s + 10·22-s − 3·23-s + 2·25-s + 24·26-s − 9·28-s + 9·29-s + 5·31-s − 6·32-s + 6·34-s − 12·35-s − 9·36-s − 10·37-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 1.78·5-s − 1.13·7-s − 1.41·8-s − 9-s − 2.52·10-s − 1.50·11-s − 3.32·13-s + 1.60·14-s + 5/4·16-s − 0.727·17-s + 1.41·18-s − 1.14·19-s + 2.68·20-s + 2.13·22-s − 0.625·23-s + 2/5·25-s + 4.70·26-s − 1.70·28-s + 1.67·29-s + 0.898·31-s − 1.06·32-s + 1.02·34-s − 2.02·35-s − 3/2·36-s − 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08370164942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08370164942\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 37 | $C_2$ | \( 1 + 10 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 5 T - 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 9 T + 10 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 3 T - 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 13 T + 80 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 9 T - 16 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−10.60127180452483226109287758507, −10.14669342912969410423228494527, −10.13478597845929654311090846893, −9.338736954203543476908025046707, −9.135643872225903641405060754848, −8.917127418175410158816818506587, −8.075640738653600967334625506050, −7.76040492271418404732183454569, −7.32298086765906183437868220752, −6.80772175267175247377015328480, −6.31503605074537273751086119122, −5.80196203996756332431567737315, −5.71888602822005666077546007186, −4.79064233170391487737135603370, −4.55469866611234894882417857756, −3.03047842943112928674304291412, −2.64166718718762584739316212858, −2.40817322406110800871618383279, −1.94552671258862664529671784722, −0.17138029874370126223161956722,
0.17138029874370126223161956722, 1.94552671258862664529671784722, 2.40817322406110800871618383279, 2.64166718718762584739316212858, 3.03047842943112928674304291412, 4.55469866611234894882417857756, 4.79064233170391487737135603370, 5.71888602822005666077546007186, 5.80196203996756332431567737315, 6.31503605074537273751086119122, 6.80772175267175247377015328480, 7.32298086765906183437868220752, 7.76040492271418404732183454569, 8.075640738653600967334625506050, 8.917127418175410158816818506587, 9.135643872225903641405060754848, 9.338736954203543476908025046707, 10.13478597845929654311090846893, 10.14669342912969410423228494527, 10.60127180452483226109287758507
