| L(s) = 1 | − 6·3-s + 11·4-s + 18·5-s + 27·9-s − 66·12-s − 108·15-s + 57·16-s + 198·20-s + 180·23-s − 7·25-s − 108·27-s − 376·31-s + 297·36-s + 266·37-s + 486·45-s + 144·47-s − 342·48-s − 98·49-s − 90·53-s + 756·59-s − 1.18e3·60-s − 77·64-s − 772·67-s − 1.08e3·69-s − 396·71-s + 42·75-s + 1.02e3·80-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 11/8·4-s + 1.60·5-s + 9-s − 1.58·12-s − 1.85·15-s + 0.890·16-s + 2.21·20-s + 1.63·23-s − 0.0559·25-s − 0.769·27-s − 2.17·31-s + 11/8·36-s + 1.18·37-s + 1.60·45-s + 0.446·47-s − 1.02·48-s − 2/7·49-s − 0.233·53-s + 1.66·59-s − 2.55·60-s − 0.150·64-s − 1.40·67-s − 1.88·69-s − 0.661·71-s + 0.0646·75-s + 1.43·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.560768966\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.560768966\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 11 T^{2} + p^{6} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 9 T + p^{3} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 p^{2} T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 649 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 7 p^{2} T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 7450 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 90 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 40975 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 188 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 133 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 136519 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 153722 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 72 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 45 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 378 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 65162 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 386 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 198 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 772226 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 962846 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 411626 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 45 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 89 T + p^{3} T^{2} )^{2} \) |
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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
−11.41505408062244613497981974528, −10.64730834709451333114530038340, −10.56939826390575742678670119827, −10.01599795453316482997661429344, −9.386983654753553971010744110187, −9.272713350031705677718181896145, −8.500437952763112088089747877112, −7.61318835890389105509109530079, −7.26151463735823526623255242708, −6.89723074268839939536178584405, −6.29664924774416492909103212221, −5.87109790700174915599183011078, −5.66920484391210539756562407247, −5.08815290414079960065966673380, −4.39417990513750377918682885445, −3.47774361089771024229608245461, −2.71981149544223796803739153556, −1.95681424528205448762640250600, −1.66267948654698761608896529334, −0.68522241447523316514750693098,
0.68522241447523316514750693098, 1.66267948654698761608896529334, 1.95681424528205448762640250600, 2.71981149544223796803739153556, 3.47774361089771024229608245461, 4.39417990513750377918682885445, 5.08815290414079960065966673380, 5.66920484391210539756562407247, 5.87109790700174915599183011078, 6.29664924774416492909103212221, 6.89723074268839939536178584405, 7.26151463735823526623255242708, 7.61318835890389105509109530079, 8.500437952763112088089747877112, 9.272713350031705677718181896145, 9.386983654753553971010744110187, 10.01599795453316482997661429344, 10.56939826390575742678670119827, 10.64730834709451333114530038340, 11.41505408062244613497981974528
