| L(s) = 1 | + 8.19e3·4-s − 3.86e5·7-s + 6.96e7·13-s − 1.77e8·19-s + 1.22e9·25-s − 3.16e9·28-s + 5.47e9·31-s + 2.73e10·37-s − 1.54e11·43-s + 5.25e10·49-s + 5.70e11·52-s − 7.66e11·61-s − 5.49e11·64-s + 2.26e11·67-s − 1.84e12·73-s − 1.45e12·76-s − 4.08e12·79-s − 2.69e13·91-s − 2.47e13·97-s + 1.00e13·100-s − 2.40e13·103-s + 6.95e12·109-s + 3.45e13·121-s + 4.48e13·124-s + 127-s + 131-s + 6.85e13·133-s + ⋯ |
| L(s) = 1 | + 4-s − 1.24·7-s + 3.99·13-s − 0.865·19-s + 25-s − 1.24·28-s + 1.10·31-s + 1.75·37-s − 3.72·43-s + 0.542·49-s + 3.99·52-s − 1.90·61-s − 64-s + 0.306·67-s − 1.42·73-s − 0.865·76-s − 1.89·79-s − 4.96·91-s − 3.01·97-s + 100-s − 1.98·103-s + 0.397·109-s + 121-s + 1.10·124-s + 1.07·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(3.499568619\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.499568619\) |
| \(L(\frac{15}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 386569 T + p^{13} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p^{13} T^{2} + p^{26} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p^{13} T^{2} + p^{26} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p^{13} T^{2} + p^{26} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 34804217 T + p^{13} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p^{13} T^{2} + p^{26} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 231547688 T + p^{13} T^{2} )( 1 + 408943081 T + p^{13} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p^{13} T^{2} + p^{26} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p^{13} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 9862529036 T + p^{13} T^{2} )( 1 + 4383058669 T + p^{13} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 26671973339 T + p^{13} T^{2} )( 1 - 705453830 T + p^{13} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p^{13} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 77218825315 T + p^{13} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p^{13} T^{2} + p^{26} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p^{13} T^{2} + p^{26} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p^{13} T^{2} + p^{26} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 171215488093 T + p^{13} T^{2} )( 1 + 595391378401 T + p^{13} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 1380765177776 T + p^{13} T^{2} )( 1 + 1153893260515 T + p^{13} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{13} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 641895299153 T + p^{13} T^{2} )( 1 + 2490400654570 T + p^{13} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 820761767284 T + p^{13} T^{2} )( 1 + 3263778980263 T + p^{13} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p^{13} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p^{13} T^{2} + p^{26} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 12352800515314 T + p^{13} T^{2} )^{2} \) |
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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
−12.67956915941264042634389961766, −11.65328818993880876206301146448, −11.48789085475342258635516206842, −10.68213127899168488907632124240, −10.63934882970677437754657853253, −9.731769791368634098701896177409, −9.005234992431773146151721747778, −8.348813345455197391641813408543, −8.225956807299065564095212428811, −6.75135494445845633567326798531, −6.72552584883798513417777713216, −6.11251382945101435924016658500, −5.80978669543641176079218593341, −4.46999659229369451296737921040, −3.86781653650923235723636543955, −3.05031666023229871851106322309, −2.94656717265091497130378969266, −1.51178987584367941609140241200, −1.43669686895672051935070492477, −0.45338920763888235954615480699,
0.45338920763888235954615480699, 1.43669686895672051935070492477, 1.51178987584367941609140241200, 2.94656717265091497130378969266, 3.05031666023229871851106322309, 3.86781653650923235723636543955, 4.46999659229369451296737921040, 5.80978669543641176079218593341, 6.11251382945101435924016658500, 6.72552584883798513417777713216, 6.75135494445845633567326798531, 8.225956807299065564095212428811, 8.348813345455197391641813408543, 9.005234992431773146151721747778, 9.731769791368634098701896177409, 10.63934882970677437754657853253, 10.68213127899168488907632124240, 11.48789085475342258635516206842, 11.65328818993880876206301146448, 12.67956915941264042634389961766
