| L(s) = 1 | + 2-s − 6·5-s + 5·7-s − 8-s − 6·10-s + 4·13-s + 5·14-s − 16-s + 6·17-s + 4·19-s − 12·23-s + 17·25-s + 4·26-s − 3·29-s − 8·31-s + 6·34-s − 30·35-s − 8·37-s + 4·38-s + 6·40-s − 6·41-s − 8·43-s − 12·46-s + 6·47-s + 18·49-s + 17·50-s + 9·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 2.68·5-s + 1.88·7-s − 0.353·8-s − 1.89·10-s + 1.10·13-s + 1.33·14-s − 1/4·16-s + 1.45·17-s + 0.917·19-s − 2.50·23-s + 17/5·25-s + 0.784·26-s − 0.557·29-s − 1.43·31-s + 1.02·34-s − 5.07·35-s − 1.31·37-s + 0.648·38-s + 0.948·40-s − 0.937·41-s − 1.21·43-s − 1.76·46-s + 0.875·47-s + 18/7·49-s + 2.40·50-s + 1.23·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.678637509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.678637509\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−10.24474279859748350997775765370, −9.690847473478284191596286237155, −8.850965196213704117003188207972, −8.521086023789416090339400667213, −8.363720281403252582957515206841, −7.937428228047051312992757881004, −7.58669821364558233895931740878, −7.30646063046583060165180985034, −6.99262262871431262151687457843, −5.90351526504624489081906013988, −5.75634757857344306268088997862, −5.17280300791767053243512504585, −4.82229648817581861426271714697, −4.09507597795701589724998820085, −3.99682078638867322083682609692, −3.44831225174260494937688310190, −3.39567322456368141208455757504, −2.08033819687691849851095425972, −1.50956903677584336550849504003, −0.53750337517195478273435520502,
0.53750337517195478273435520502, 1.50956903677584336550849504003, 2.08033819687691849851095425972, 3.39567322456368141208455757504, 3.44831225174260494937688310190, 3.99682078638867322083682609692, 4.09507597795701589724998820085, 4.82229648817581861426271714697, 5.17280300791767053243512504585, 5.75634757857344306268088997862, 5.90351526504624489081906013988, 6.99262262871431262151687457843, 7.30646063046583060165180985034, 7.58669821364558233895931740878, 7.937428228047051312992757881004, 8.363720281403252582957515206841, 8.521086023789416090339400667213, 8.850965196213704117003188207972, 9.690847473478284191596286237155, 10.24474279859748350997775765370
