| L(s) = 1 | − 64·4-s − 729·9-s − 120·11-s + 4.09e3·16-s − 1.07e5·19-s + 1.77e5·29-s − 2.40e5·31-s + 4.66e4·36-s − 1.02e6·41-s + 7.68e3·44-s + 1.59e6·49-s + 3.77e6·59-s − 5.43e5·61-s − 2.62e5·64-s − 5.01e6·71-s + 6.85e6·76-s + 1.47e7·79-s + 5.31e5·81-s − 1.13e7·89-s + 8.74e4·99-s + 3.09e7·101-s − 2.79e7·109-s − 1.13e7·116-s − 3.89e7·121-s + 1.54e7·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s − 0.0271·11-s + 1/4·16-s − 3.58·19-s + 1.34·29-s − 1.45·31-s + 1/6·36-s − 2.31·41-s + 0.0135·44-s + 1.93·49-s + 2.39·59-s − 0.306·61-s − 1/8·64-s − 1.66·71-s + 1.79·76-s + 3.35·79-s + 1/9·81-s − 1.70·89-s + 0.00906·99-s + 2.98·101-s − 2.06·109-s − 0.674·116-s − 1.99·121-s + 0.725·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.04511486412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04511486412\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2$ | \( 1 + p^{6} T^{2} \) |
| 3 | $C_2$ | \( 1 + p^{6} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 1593262 T^{2} + p^{14} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 60 T + p^{7} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 72017882 T^{2} + p^{14} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 626409502 T^{2} + p^{14} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 53564 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6080650894 T^{2} + p^{14} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 88554 T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 120352 T + p^{7} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 30703611050 T^{2} + p^{14} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 510246 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 522492572470 T^{2} + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 416699614 p^{2} T^{2} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2340511297270 T^{2} + p^{14} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 1885740 T + p^{7} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 271690 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 10966511900422 T^{2} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2505480 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 13365336789838 T^{2} + p^{14} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 7354768 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 39474570255718 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 5685162 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 97894684984510 T^{2} + p^{14} T^{4} \) |
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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
−11.98278587301169367229765565053, −11.47377464616022301050744821196, −10.74783962696890619061605247948, −10.36971350999318702306159097420, −10.18855390185646514100500907918, −9.173077592368046417606235682767, −8.643659432951926488431297602800, −8.608561267472088070993384310798, −7.914274395369269704113067026260, −7.10502438244692577174599222356, −6.48708723227243605919253497021, −6.18347946927772991782764127424, −5.26387778019591580998147894678, −4.83698977910186591106787327839, −3.95050769453963841946478366774, −3.76985407135732910749375646218, −2.52199166568518671393580452347, −2.15007180517046887256470830912, −1.18153489576986631782379282170, −0.06064438411723357706532854012,
0.06064438411723357706532854012, 1.18153489576986631782379282170, 2.15007180517046887256470830912, 2.52199166568518671393580452347, 3.76985407135732910749375646218, 3.95050769453963841946478366774, 4.83698977910186591106787327839, 5.26387778019591580998147894678, 6.18347946927772991782764127424, 6.48708723227243605919253497021, 7.10502438244692577174599222356, 7.914274395369269704113067026260, 8.608561267472088070993384310798, 8.643659432951926488431297602800, 9.173077592368046417606235682767, 10.18855390185646514100500907918, 10.36971350999318702306159097420, 10.74783962696890619061605247948, 11.47377464616022301050744821196, 11.98278587301169367229765565053
