| L(s) = 1 | + 90·5-s + 90·9-s + 504·11-s − 440·19-s + 4.97e3·25-s + 1.38e4·29-s − 1.35e4·31-s − 396·41-s + 8.10e3·45-s + 3.00e4·49-s + 4.53e4·55-s − 4.93e4·59-s + 1.13e4·61-s − 1.06e5·71-s − 1.03e5·79-s − 5.09e4·81-s − 1.99e4·89-s − 3.96e4·95-s + 4.53e4·99-s + 2.18e5·101-s + 4.20e4·109-s − 1.31e5·121-s + 1.66e5·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1.60·5-s + 0.370·9-s + 1.25·11-s − 0.279·19-s + 1.59·25-s + 3.06·29-s − 2.52·31-s − 0.0367·41-s + 0.596·45-s + 1.78·49-s + 2.02·55-s − 1.84·59-s + 0.392·61-s − 2.51·71-s − 1.87·79-s − 0.862·81-s − 0.267·89-s − 0.450·95-s + 0.465·99-s + 2.12·101-s + 0.338·109-s − 0.817·121-s + 0.953·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.206347680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.206347680\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 18 p T + p^{5} T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 8 p T + p^{5} T^{2} )( 1 + 8 p T + p^{5} T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 30050 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 252 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 728330 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2363810 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 220 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6946370 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6930 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6752 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 56462470 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 198 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 293842250 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 347593490 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 802472090 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 24660 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 5698 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 795787610 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 53352 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 883886830 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 51920 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 4053674810 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 9990 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 6923133890 T^{2} + p^{10} T^{4} \) |
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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
−10.80201467164829489701184853755, −10.40892858770794560736968889890, −10.23730349310697508008336373242, −9.565311661586334235127803908271, −9.229959430750911535329708823254, −8.644970969501270960808861750581, −8.616892640421080972514119794961, −7.42688553001147750088325397244, −7.21659796138642341754144825226, −6.50809377868913142067125706158, −6.19374378876673960503909697346, −5.73446475358589556444028298708, −5.12455940119600115867489365799, −4.46279834610613191267000440772, −4.02121513750717320755211096792, −3.09033000781569792967789973524, −2.59172859862248909777134599662, −1.65943898943051853254038489307, −1.47481374772971351464535761853, −0.60490808402206526809814464569,
0.60490808402206526809814464569, 1.47481374772971351464535761853, 1.65943898943051853254038489307, 2.59172859862248909777134599662, 3.09033000781569792967789973524, 4.02121513750717320755211096792, 4.46279834610613191267000440772, 5.12455940119600115867489365799, 5.73446475358589556444028298708, 6.19374378876673960503909697346, 6.50809377868913142067125706158, 7.21659796138642341754144825226, 7.42688553001147750088325397244, 8.616892640421080972514119794961, 8.644970969501270960808861750581, 9.229959430750911535329708823254, 9.565311661586334235127803908271, 10.23730349310697508008336373242, 10.40892858770794560736968889890, 10.80201467164829489701184853755
