| L(s) = 1 | − 4·4-s − 10·9-s + 136·11-s + 16·16-s + 256·19-s − 572·29-s − 48·31-s + 40·36-s + 132·41-s − 544·44-s − 49·49-s − 336·59-s + 340·61-s − 64·64-s + 1.23e3·71-s − 1.02e3·76-s + 1.88e3·79-s − 629·81-s + 2.86e3·89-s − 1.36e3·99-s + 548·109-s + 2.28e3·116-s + 1.12e4·121-s + 192·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.370·9-s + 3.72·11-s + 1/4·16-s + 3.09·19-s − 3.66·29-s − 0.278·31-s + 5/27·36-s + 0.502·41-s − 1.86·44-s − 1/7·49-s − 0.741·59-s + 0.713·61-s − 1/8·64-s + 2.05·71-s − 1.54·76-s + 2.68·79-s − 0.862·81-s + 3.40·89-s − 1.38·99-s + 0.481·109-s + 1.83·116-s + 8.42·121-s + 0.139·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.338974255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.338974255\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 10 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 68 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3238 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4350 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 128 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 17934 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 286 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 24 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 14870 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 66 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 143638 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 110302 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 296598 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 168 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 170 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 283430 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 616 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 715534 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 944 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 691990 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1430 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 212446 T^{2} + p^{6} T^{4} \) |
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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
−11.37758674891931112752530199051, −11.19772956073483182787878321825, −10.22246278775703251686411453023, −9.537448279857952871734424653719, −9.402693801341628379748715644920, −9.023493192182608778269263891818, −8.978593265573081419950257119560, −7.71342282258077603476216341656, −7.65589050851761258825871757975, −7.03153698065915145625887608989, −6.38307919564557757871205388502, −6.05494612550693077171608512909, −5.32694958686946087110719599442, −4.96884031490325487724974862433, −3.83537472411058077126168625249, −3.73788865078859095381703028553, −3.41416618816019949659777624879, −1.99768619795964406637564156874, −1.30998528449129112665768201297, −0.75608890385025475783059446804,
0.75608890385025475783059446804, 1.30998528449129112665768201297, 1.99768619795964406637564156874, 3.41416618816019949659777624879, 3.73788865078859095381703028553, 3.83537472411058077126168625249, 4.96884031490325487724974862433, 5.32694958686946087110719599442, 6.05494612550693077171608512909, 6.38307919564557757871205388502, 7.03153698065915145625887608989, 7.65589050851761258825871757975, 7.71342282258077603476216341656, 8.978593265573081419950257119560, 9.023493192182608778269263891818, 9.402693801341628379748715644920, 9.537448279857952871734424653719, 10.22246278775703251686411453023, 11.19772956073483182787878321825, 11.37758674891931112752530199051
