L(s) = 1 | − 16·4-s + 485·9-s − 242·11-s + 256·16-s − 2.84e3·19-s + 4.12e3·29-s + 1.24e4·31-s − 7.76e3·36-s + 1.06e4·41-s + 3.87e3·44-s + 6.05e3·49-s + 6.99e4·59-s − 9.18e4·61-s − 4.09e3·64-s + 2.66e4·71-s + 4.55e4·76-s − 1.54e5·79-s + 1.76e5·81-s − 2.50e5·89-s − 1.17e5·99-s + 2.96e3·101-s − 1.75e5·109-s − 6.60e4·116-s + 4.39e4·121-s − 1.99e5·124-s + 127-s + 131-s + ⋯ |
L(s) = 1 | − 1/2·4-s + 1.99·9-s − 0.603·11-s + 1/4·16-s − 1.80·19-s + 0.911·29-s + 2.33·31-s − 0.997·36-s + 0.985·41-s + 0.301·44-s + 0.360·49-s + 2.61·59-s − 3.16·61-s − 1/8·64-s + 0.626·71-s + 0.904·76-s − 2.78·79-s + 2.98·81-s − 3.35·89-s − 1.20·99-s + 0.0289·101-s − 1.41·109-s − 0.455·116-s + 3/11·121-s − 1.16·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 302500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 302500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.027700855\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.027700855\) |
\(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
good | 3 | $C_2^2$ | \( 1 - 485 T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6058 T^{2} + p^{10} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 263722 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2295070 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 1424 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 9707845 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2064 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6245 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 79908311 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 5304 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 22751918 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 163400158 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 107696090 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 34989 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 45940 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2057982565 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 13311 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 1309515586 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 77234 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 4851541090 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 125415 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 9287997265 T^{2} + p^{10} T^{4} \) |
show more | | |
show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.26772840484136251331668946053, −9.902828077097590753236647029353, −9.503289496218227891152006339862, −8.690529889178830796639051056589, −8.616530177468983034762567960627, −7.973086720598388884943927681493, −7.62282010702504340294586630868, −7.03989666314006893544576883718, −6.61906687308860739010336821652, −6.24525504963952091761986971035, −5.62666795124047377613043516883, −4.89541214036486517515099927716, −4.47206890521511149385129833377, −4.24859334485509548506669224700, −3.76786314485316018272266988607, −2.67726972979210351509070079934, −2.52412045818749208559136438371, −1.47190582118652475489307462558, −1.17284060619214362295062690224, −0.34790198358547525806986581996,
0.34790198358547525806986581996, 1.17284060619214362295062690224, 1.47190582118652475489307462558, 2.52412045818749208559136438371, 2.67726972979210351509070079934, 3.76786314485316018272266988607, 4.24859334485509548506669224700, 4.47206890521511149385129833377, 4.89541214036486517515099927716, 5.62666795124047377613043516883, 6.24525504963952091761986971035, 6.61906687308860739010336821652, 7.03989666314006893544576883718, 7.62282010702504340294586630868, 7.973086720598388884943927681493, 8.616530177468983034762567960627, 8.690529889178830796639051056589, 9.503289496218227891152006339862, 9.902828077097590753236647029353, 10.26772840484136251331668946053