| L(s) = 1 | + 64·4-s + 361·9-s − 1.14e3·11-s + 3.07e3·16-s − 3.12e3·25-s − 1.76e4·29-s + 2.31e4·36-s − 7.33e4·44-s + 1.31e5·64-s + 1.24e5·71-s + 6.16e4·79-s + 7.12e4·81-s − 4.13e5·99-s − 2.00e5·100-s + 1.78e5·109-s − 1.12e6·116-s + 6.62e5·121-s + 127-s + 131-s + 137-s + 139-s + 1.10e6·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 2·4-s + 1.48·9-s − 2.85·11-s + 3·16-s − 25-s − 3.89·29-s + 2.97·36-s − 5.71·44-s + 4·64-s + 2.92·71-s + 1.11·79-s + 1.20·81-s − 4.24·99-s − 2·100-s + 1.43·109-s − 7.78·116-s + 4.11·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 4.45·144-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.486059739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.486059739\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 + p^{5} T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - 361 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 573 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 606461 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2620411 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 8811 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 455938111 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 62148 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 3825881314 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 30811 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 3202399286 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 16228370611 T^{2} + p^{10} T^{4} \) |
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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
−11.27461623024188663544383971568, −10.92060395418865175526790916014, −10.71675306597216730077220265473, −10.06374027180453762594496694746, −9.864744878905794176705464288648, −9.249955224563126587120124456851, −8.042368638644076454258379805265, −7.898796190018977891408489645627, −7.46463331656068543992402273611, −7.19847247766966711303109755098, −6.56491433804549542488108664658, −5.75747311582998373778903550290, −5.52093533578797804916728539247, −4.94457909520986410898648410789, −3.81352533573842967548253362415, −3.42241523994757601492439545733, −2.46261793636966227566537894526, −2.10178069346125691851133084469, −1.64884607325033885425645097539, −0.47631037556779073456721581429,
0.47631037556779073456721581429, 1.64884607325033885425645097539, 2.10178069346125691851133084469, 2.46261793636966227566537894526, 3.42241523994757601492439545733, 3.81352533573842967548253362415, 4.94457909520986410898648410789, 5.52093533578797804916728539247, 5.75747311582998373778903550290, 6.56491433804549542488108664658, 7.19847247766966711303109755098, 7.46463331656068543992402273611, 7.898796190018977891408489645627, 8.042368638644076454258379805265, 9.249955224563126587120124456851, 9.864744878905794176705464288648, 10.06374027180453762594496694746, 10.71675306597216730077220265473, 10.92060395418865175526790916014, 11.27461623024188663544383971568
