| L(s) = 1 | − 16·4-s − 1.55e3·11-s + 256·16-s − 2.29e3·19-s − 9.84e3·29-s + 3.60e3·31-s + 3.02e4·41-s + 2.48e4·44-s + 1.34e4·49-s + 6.79e4·59-s + 9.48e4·61-s − 4.09e3·64-s + 1.50e4·71-s + 3.66e4·76-s − 1.51e5·79-s − 6.11e4·89-s + 4.77e4·101-s + 1.41e4·109-s + 1.57e5·116-s + 1.48e6·121-s − 5.76e4·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 3.87·11-s + 1/4·16-s − 1.45·19-s − 2.17·29-s + 0.673·31-s + 2.81·41-s + 1.93·44-s + 0.800·49-s + 2.54·59-s + 3.26·61-s − 1/8·64-s + 0.355·71-s + 0.727·76-s − 2.73·79-s − 0.818·89-s + 0.466·101-s + 0.114·109-s + 1.08·116-s + 9.24·121-s − 0.336·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4817824544\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4817824544\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 13450 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 777 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 230 p^{2} T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2838985 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 1145 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 9435370 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4920 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 1802 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 34971770 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 15123 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 232488550 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 413370190 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 824735590 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 33960 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 47402 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2526616885 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 7548 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 567591145 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 75830 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 5734483885 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 30585 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 6354936190 T^{2} + p^{10} T^{4} \) |
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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
−10.39798964381617813173956519941, −10.08070785412102863859938006182, −9.876930832172200695011874119753, −9.113309266240271976704600901256, −8.447017316927434724124767612853, −8.378929551534074110360284887203, −7.79677875544800331994117195032, −7.36679159427633730011517240957, −7.10326703340955306835827102673, −6.00151404085928842636337604468, −5.77393616665010670710115758991, −5.13613098268521036212728956675, −5.07758870351346745074654564744, −4.08447346433326208727657147252, −3.87573938792382764920838421590, −2.66615829984307915335379565365, −2.57227077125047075810360687616, −2.08720756976912269066861236834, −0.805063086791252708433563382570, −0.21461916645196355919668462331,
0.21461916645196355919668462331, 0.805063086791252708433563382570, 2.08720756976912269066861236834, 2.57227077125047075810360687616, 2.66615829984307915335379565365, 3.87573938792382764920838421590, 4.08447346433326208727657147252, 5.07758870351346745074654564744, 5.13613098268521036212728956675, 5.77393616665010670710115758991, 6.00151404085928842636337604468, 7.10326703340955306835827102673, 7.36679159427633730011517240957, 7.79677875544800331994117195032, 8.378929551534074110360284887203, 8.447017316927434724124767612853, 9.113309266240271976704600901256, 9.876930832172200695011874119753, 10.08070785412102863859938006182, 10.39798964381617813173956519941
