| L(s) = 1 | − 16·4-s − 190·9-s − 1.53e3·11-s + 256·16-s − 2.20e3·19-s + 1.12e4·29-s − 7.97e3·31-s + 3.04e3·36-s + 3.08e3·41-s + 2.45e4·44-s + 3.31e4·49-s − 5.67e4·59-s + 1.10e4·61-s − 4.09e3·64-s + 8.47e4·71-s + 3.52e4·76-s + 7.92e4·79-s − 2.29e4·81-s − 1.15e5·89-s + 2.91e5·99-s − 2.82e5·101-s − 4.36e5·109-s − 1.79e5·116-s + 1.44e6·121-s + 1.27e5·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.781·9-s − 3.82·11-s + 1/4·16-s − 1.39·19-s + 2.47·29-s − 1.49·31-s + 0.390·36-s + 0.286·41-s + 1.91·44-s + 1.97·49-s − 2.12·59-s + 0.380·61-s − 1/8·64-s + 1.99·71-s + 0.699·76-s + 1.42·79-s − 0.388·81-s − 1.54·89-s + 2.99·99-s − 2.75·101-s − 3.52·109-s − 1.23·116-s + 8.98·121-s + 0.745·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4051008259\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4051008259\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 190 T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 33130 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 768 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 740470 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2696830 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 1100 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 8928490 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5610 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 3988 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 138667750 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 1542 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 268756210 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 153278630 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 635715430 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 28380 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 5522 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2088083650 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 42372 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 1429023310 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 39640 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 4298931010 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 57690 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 3671481410 T^{2} + p^{10} T^{4} \) |
| 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
−15.23501730075735007855910143481, −13.93606359404150330918530790071, −13.82250123480464341421603471862, −13.05265918477547073785973615812, −12.64880755379914409746304691856, −12.22008494901334422419986718194, −10.99876659575571777670651588653, −10.54250197546764072898436812530, −10.46572395375783880175663740848, −9.445664976751786812730331439591, −8.539798430126657057518600147150, −8.118908283399359007773267981675, −7.69996137015797408150125367913, −6.60999869451165999819427217355, −5.45831892377739789670431493775, −5.30505788922073790936057385707, −4.32854548765431288603696569022, −2.86332319494168271911263162703, −2.44318283480698854920959325336, −0.30827538427490138079739408299,
0.30827538427490138079739408299, 2.44318283480698854920959325336, 2.86332319494168271911263162703, 4.32854548765431288603696569022, 5.30505788922073790936057385707, 5.45831892377739789670431493775, 6.60999869451165999819427217355, 7.69996137015797408150125367913, 8.118908283399359007773267981675, 8.539798430126657057518600147150, 9.445664976751786812730331439591, 10.46572395375783880175663740848, 10.54250197546764072898436812530, 10.99876659575571777670651588653, 12.22008494901334422419986718194, 12.64880755379914409746304691856, 13.05265918477547073785973615812, 13.82250123480464341421603471862, 13.93606359404150330918530790071, 15.23501730075735007855910143481
