| L(s) = 1 | − 4·4-s + 60·11-s + 16·16-s + 98·19-s − 372·29-s − 320·31-s − 756·41-s − 240·44-s + 517·49-s + 408·59-s − 1.75e3·61-s − 64·64-s + 1.21e3·71-s − 392·76-s − 2.30e3·79-s + 1.96e3·89-s + 2.49e3·101-s − 4.39e3·109-s + 1.48e3·116-s + 38·121-s + 1.28e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.64·11-s + 1/4·16-s + 1.18·19-s − 2.38·29-s − 1.85·31-s − 2.87·41-s − 0.822·44-s + 1.50·49-s + 0.900·59-s − 3.68·61-s − 1/8·64-s + 2.02·71-s − 0.591·76-s − 3.27·79-s + 2.34·89-s + 2.45·101-s − 3.86·109-s + 1.19·116-s + 0.0285·121-s + 0.926·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9896481505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9896481505\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 517 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 30 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 673 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9682 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 49 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 24010 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 186 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 160 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 93025 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 378 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 87190 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 186910 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 27146 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 204 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 877 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 566557 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 606 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 592273 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 1151 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1133170 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 984 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1755121 T^{2} + p^{6} T^{4} \) |
| 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
−9.278196941876454424326107901711, −9.004440073690009238713780197191, −8.967069346701478006011235648731, −8.283132753826826760982727541983, −7.69059077828809909688700807896, −7.45539383893865934755729516213, −7.03506615574261596947264000540, −6.63929730654231367033701296876, −6.07659504837577410072065228084, −5.66269033410536037170703666193, −5.23765791618369901198248961824, −4.93447108708793348394713127346, −4.05935219333665362592150762020, −3.94477925098019763418744864850, −3.41038932048985069281587079859, −3.07132678374584473819179441639, −2.00453666748180950143322196512, −1.65123767115518037018487535282, −1.15653420751520522632989679516, −0.24154588075604833752331488406,
0.24154588075604833752331488406, 1.15653420751520522632989679516, 1.65123767115518037018487535282, 2.00453666748180950143322196512, 3.07132678374584473819179441639, 3.41038932048985069281587079859, 3.94477925098019763418744864850, 4.05935219333665362592150762020, 4.93447108708793348394713127346, 5.23765791618369901198248961824, 5.66269033410536037170703666193, 6.07659504837577410072065228084, 6.63929730654231367033701296876, 7.03506615574261596947264000540, 7.45539383893865934755729516213, 7.69059077828809909688700807896, 8.283132753826826760982727541983, 8.967069346701478006011235648731, 9.004440073690009238713780197191, 9.278196941876454424326107901711
