| L(s) = 1 | − 9·9-s − 72·11-s + 56·19-s + 372·29-s + 352·31-s − 60·41-s − 49·49-s + 1.12e3·59-s − 644·61-s − 96·71-s + 992·79-s + 81·81-s − 2.62e3·89-s + 648·99-s + 2.07e3·101-s + 1.07e3·109-s + 1.22e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.23e3·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 1.97·11-s + 0.676·19-s + 2.38·29-s + 2.03·31-s − 0.228·41-s − 1/7·49-s + 2.48·59-s − 1.35·61-s − 0.160·71-s + 1.41·79-s + 1/9·81-s − 3.12·89-s + 0.657·99-s + 2.04·101-s + 0.945·109-s + 0.921·121-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 + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.47·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.239348776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.239348776\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + 36 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3238 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9790 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 28 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 12530 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 186 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 176 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 73418 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 30 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 10730 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 21022 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 204118 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 564 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 322 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 88870 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 48 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 384050 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 496 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 924550 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1314 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 242498 T^{2} + p^{6} T^{4} \) |
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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
−8.734929431542620152009032501132, −8.386176837901720016061457975226, −8.272576904745783059630702332566, −7.935202350023116594696980665213, −7.33748571211491957759774723135, −7.13587130067315372046697426745, −6.47326311175567967049811749956, −6.31150237624100314912831340859, −5.64611996167613717596548255451, −5.42459525683894264807727473101, −4.78683324838058955662225251178, −4.74639242893538630970878805097, −4.16513354780198424437595825245, −3.43150032102024509899743071341, −2.90715903213987040638554415528, −2.74988301121841387995468060550, −2.28627856871080200591304076020, −1.52297557768678847880597076959, −0.70648461417939153523242630146, −0.53662744540970770279638032834,
0.53662744540970770279638032834, 0.70648461417939153523242630146, 1.52297557768678847880597076959, 2.28627856871080200591304076020, 2.74988301121841387995468060550, 2.90715903213987040638554415528, 3.43150032102024509899743071341, 4.16513354780198424437595825245, 4.74639242893538630970878805097, 4.78683324838058955662225251178, 5.42459525683894264807727473101, 5.64611996167613717596548255451, 6.31150237624100314912831340859, 6.47326311175567967049811749956, 7.13587130067315372046697426745, 7.33748571211491957759774723135, 7.935202350023116594696980665213, 8.272576904745783059630702332566, 8.386176837901720016061457975226, 8.734929431542620152009032501132
