| L(s) = 1 | − 9·4-s − 9·9-s − 22·11-s + 17·16-s − 94·19-s + 108·29-s + 356·31-s + 81·36-s + 278·41-s + 198·44-s + 677·49-s + 1.25e3·59-s + 640·61-s + 423·64-s − 1.89e3·71-s + 846·76-s + 1.44e3·79-s + 81·81-s − 808·89-s + 198·99-s − 1.09e3·101-s + 1.15e3·109-s − 972·116-s + 363·121-s − 3.20e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 9/8·4-s − 1/3·9-s − 0.603·11-s + 0.265·16-s − 1.13·19-s + 0.691·29-s + 2.06·31-s + 3/8·36-s + 1.05·41-s + 0.678·44-s + 1.97·49-s + 2.75·59-s + 1.34·61-s + 0.826·64-s − 3.16·71-s + 1.27·76-s + 2.05·79-s + 1/9·81-s − 0.962·89-s + 0.201·99-s − 1.07·101-s + 1.01·109-s − 0.777·116-s + 3/11·121-s − 2.32·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.989827569\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.989827569\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 9 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 677 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3370 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8737 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 47 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 11565 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 54 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 178 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 100945 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 139 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 64150 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 169621 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 274650 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 625 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 320 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 561526 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 947 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 577330 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 721 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1123410 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 404 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1819105 T^{2} + p^{6} T^{4} \) |
| 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
−9.842417329751743445817387824533, −9.841139287661426710788325195885, −9.090754077471905503795104381851, −8.699171779132760610936321761552, −8.360905321155435004219060395628, −8.252021314928925186019479405980, −7.51150433612711306487467253938, −7.04892339545878997809637865298, −6.58501267153422060534714171772, −6.02899905503742574519540327488, −5.61325758349549965541769990694, −5.12592780607657543352540847346, −4.60620891709721076657122269766, −4.16645588046454877189034600445, −3.91584705611531397102102537421, −2.91973842983730435902445869112, −2.60213957766508378798122485212, −1.94195649750244010518997734865, −0.73231626125972804485249840533, −0.58959185299038811259206839098,
0.58959185299038811259206839098, 0.73231626125972804485249840533, 1.94195649750244010518997734865, 2.60213957766508378798122485212, 2.91973842983730435902445869112, 3.91584705611531397102102537421, 4.16645588046454877189034600445, 4.60620891709721076657122269766, 5.12592780607657543352540847346, 5.61325758349549965541769990694, 6.02899905503742574519540327488, 6.58501267153422060534714171772, 7.04892339545878997809637865298, 7.51150433612711306487467253938, 8.252021314928925186019479405980, 8.360905321155435004219060395628, 8.699171779132760610936321761552, 9.090754077471905503795104381851, 9.841139287661426710788325195885, 9.842417329751743445817387824533
