| L(s) = 1 | − 9·9-s + 8·11-s − 184·19-s + 212·29-s − 288·31-s − 780·41-s − 49·49-s + 728·59-s + 1.35e3·61-s − 16·71-s − 768·79-s + 81·81-s − 2.38e3·89-s − 72·99-s + 796·101-s + 1.87e3·109-s − 2.61e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.47e3·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 0.219·11-s − 2.22·19-s + 1.35·29-s − 1.66·31-s − 2.97·41-s − 1/7·49-s + 1.60·59-s + 2.84·61-s − 0.0267·71-s − 1.09·79-s + 1/9·81-s − 2.84·89-s − 0.0730·99-s + 0.784·101-s + 1.64·109-s − 1.96·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 0.672·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8064627762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8064627762\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 4 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 1478 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9630 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 92 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1230 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 106 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 144 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 76342 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 390 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 99050 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 71138 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 69482 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 364 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 678 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 110810 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 599950 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 384 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 843270 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1194 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 430658 T^{2} + p^{6} T^{4} \) |
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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
−8.873710292611585553740468772235, −8.542222737489697285052910572286, −8.245525450570863198102638050201, −8.014987727968908900509961073315, −7.23360785145816925042800614774, −6.86202624460057260219195230945, −6.70870919036987273729668193862, −6.32606465557654087179360601218, −5.71584198271652716078803442801, −5.34907356905472767499163831716, −5.03452139139164025086740393084, −4.45315358010991380576863299478, −3.86524136341126907758285643072, −3.85464774016382354694199052601, −3.03862808860091806089384428942, −2.64518240234891050233234077620, −1.91986023388424447428465672186, −1.78937730751274217184649329827, −0.893914474519227313754950317785, −0.20740427915651909944462589503,
0.20740427915651909944462589503, 0.893914474519227313754950317785, 1.78937730751274217184649329827, 1.91986023388424447428465672186, 2.64518240234891050233234077620, 3.03862808860091806089384428942, 3.85464774016382354694199052601, 3.86524136341126907758285643072, 4.45315358010991380576863299478, 5.03452139139164025086740393084, 5.34907356905472767499163831716, 5.71584198271652716078803442801, 6.32606465557654087179360601218, 6.70870919036987273729668193862, 6.86202624460057260219195230945, 7.23360785145816925042800614774, 8.014987727968908900509961073315, 8.245525450570863198102638050201, 8.542222737489697285052910572286, 8.873710292611585553740468772235
