| L(s) = 1 | + 16·4-s − 9·9-s − 22·11-s + 192·16-s − 244·19-s − 192·29-s − 224·31-s − 144·36-s − 192·41-s − 352·44-s + 682·49-s − 1.32e3·59-s − 860·61-s + 2.04e3·64-s + 336·71-s − 3.90e3·76-s + 1.41e3·79-s + 81·81-s + 12·89-s + 198·99-s − 1.92e3·101-s − 1.22e3·109-s − 3.07e3·116-s + 363·121-s − 3.58e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 2·4-s − 1/3·9-s − 0.603·11-s + 3·16-s − 2.94·19-s − 1.22·29-s − 1.29·31-s − 2/3·36-s − 0.731·41-s − 1.20·44-s + 1.98·49-s − 2.91·59-s − 1.80·61-s + 4·64-s + 0.561·71-s − 5.89·76-s + 2.01·79-s + 1/9·81-s + 0.0142·89-s + 0.201·99-s − 1.89·101-s − 1.07·109-s − 2.45·116-s + 3/11·121-s − 2.59·124-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
| \(L(2)\) |
\(\approx\) |
\(1.919336920\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.919336920\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 682 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3910 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4642 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 122 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 19150 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 96 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 112 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 30550 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 96 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 13090 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78046 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 p^{2} T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 660 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 430 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 457126 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 168 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 730510 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 706 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 2950 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1354750 T^{2} + p^{6} T^{4} \) |
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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
−10.50210413317770532520602311530, −9.469336496769349918367958688757, −9.413171273882073663846105758330, −8.475754078358756440594072476480, −8.465639860056790505507724692842, −7.71038867048568857713098436136, −7.50696263722078808079534927356, −7.06051459126339102881501672752, −6.51814503178192843582106513828, −6.04219673770863734576177467077, −6.03973840526434426074794779326, −5.27896774890494054766908552370, −4.79129643749256951234640483491, −3.79220746392183930403233684744, −3.76970891299790811160157394364, −2.65849254525560447036006237603, −2.63230275770641676940415652563, −1.77881667053972672895781121368, −1.64281685125503533158083451526, −0.31687964756715890919229843694,
0.31687964756715890919229843694, 1.64281685125503533158083451526, 1.77881667053972672895781121368, 2.63230275770641676940415652563, 2.65849254525560447036006237603, 3.76970891299790811160157394364, 3.79220746392183930403233684744, 4.79129643749256951234640483491, 5.27896774890494054766908552370, 6.03973840526434426074794779326, 6.04219673770863734576177467077, 6.51814503178192843582106513828, 7.06051459126339102881501672752, 7.50696263722078808079534927356, 7.71038867048568857713098436136, 8.465639860056790505507724692842, 8.475754078358756440594072476480, 9.413171273882073663846105758330, 9.469336496769349918367958688757, 10.50210413317770532520602311530
