| L(s) = 1 | + 16·4-s − 26·9-s + 144·11-s + 192·16-s − 125·25-s − 108·29-s − 416·36-s + 2.30e3·44-s + 2.04e3·64-s + 1.65e3·71-s − 472·79-s − 53·81-s − 3.74e3·99-s − 2.00e3·100-s − 4.53e3·109-s − 1.72e3·116-s + 1.28e4·121-s + 127-s + 131-s + 137-s + 139-s − 4.99e3·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 2·4-s − 0.962·9-s + 3.94·11-s + 3·16-s − 25-s − 0.691·29-s − 1.92·36-s + 7.89·44-s + 4·64-s + 2.76·71-s − 0.672·79-s − 0.0727·81-s − 3.80·99-s − 2·100-s − 3.98·109-s − 1.38·116-s + 9.68·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 2.88·144-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.628851884\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.628851884\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 + p^{3} T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + 26 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 72 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 2774 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 754 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 54 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 175646 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 828 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 504254 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 236 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 1141306 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 897874 T^{2} + p^{6} T^{4} \) |
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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
−11.72221884220099760174484972402, −11.66708176227831290739918113127, −10.97638368278578619035849240341, −10.85685632124290718639629906432, −9.872774483394725886451998677363, −9.435336568831901701373345926530, −9.162951776516776339956241364737, −8.350831308923957004675120797338, −7.991483698153697124190179689574, −7.13280214200281486851281494018, −6.67224109268958267176189158550, −6.61829920154905712611824410531, −5.83965751070568621648854570026, −5.64968119448008775649461324418, −4.28614670259157186217193503003, −3.68091501614479129128928315337, −3.32160573401672364744400192191, −2.27563373775041971368817411136, −1.63917198563046072442386877385, −1.03665664659032309756594479838,
1.03665664659032309756594479838, 1.63917198563046072442386877385, 2.27563373775041971368817411136, 3.32160573401672364744400192191, 3.68091501614479129128928315337, 4.28614670259157186217193503003, 5.64968119448008775649461324418, 5.83965751070568621648854570026, 6.61829920154905712611824410531, 6.67224109268958267176189158550, 7.13280214200281486851281494018, 7.991483698153697124190179689574, 8.350831308923957004675120797338, 9.162951776516776339956241364737, 9.435336568831901701373345926530, 9.872774483394725886451998677363, 10.85685632124290718639629906432, 10.97638368278578619035849240341, 11.66708176227831290739918113127, 11.72221884220099760174484972402
