| L(s) = 1 | − 36·4-s − 98·7-s − 1.03e3·13-s + 272·16-s − 2.96e3·19-s − 4.87e3·25-s + 3.52e3·28-s − 5.20e3·31-s + 804·37-s + 1.39e4·43-s + 7.20e3·49-s + 3.72e4·52-s − 4.52e4·61-s + 2.70e4·64-s − 2.62e4·67-s − 1.65e5·73-s + 1.06e5·76-s − 1.62e5·79-s + 1.01e5·91-s − 2.12e4·97-s + 1.75e5·100-s − 1.51e5·103-s + 3.25e5·109-s − 2.66e4·112-s − 2.70e5·121-s + 1.87e5·124-s + 127-s + ⋯ |
| L(s) = 1 | − 9/8·4-s − 0.755·7-s − 1.70·13-s + 0.265·16-s − 1.88·19-s − 1.56·25-s + 0.850·28-s − 0.973·31-s + 0.0965·37-s + 1.14·43-s + 3/7·49-s + 1.91·52-s − 1.55·61-s + 0.826·64-s − 0.714·67-s − 3.63·73-s + 2.12·76-s − 2.92·79-s + 1.28·91-s − 0.229·97-s + 1.75·100-s − 1.41·103-s + 2.62·109-s − 0.200·112-s − 1.67·121-s + 1.09·124-s + 5.50e−6·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 9 p^{2} T^{2} + p^{10} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4878 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 270330 T^{2} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 518 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2234662 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 1484 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1763726 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 41011098 T^{2} + p^{10} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 84 p T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 402 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 231315894 T^{2} + p^{10} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 6956 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 288561554 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 94518182 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 605546330 T^{2} + p^{10} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 22610 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 13124 T + p^{5} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 1284169714 T^{2} + p^{10} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 82866 T + p^{5} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 81112 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 3432801286 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 4888727754 T^{2} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10626 T + p^{5} T^{2} )^{2} \) |
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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
−13.49228990439729328905237985585, −13.24496901716958515000112480750, −12.56671209265015014204324459688, −12.29232338348535209439525579197, −11.48864531792427613986059689074, −10.70874564856168831189094202678, −9.987057032490220392773635664328, −9.734992687492277956512636905242, −8.949886996552283651121507945444, −8.630678699026020291422832201653, −7.58447148639968036921163437354, −7.17900529342630246869617479733, −6.12829248980041301724103146750, −5.58225842583335733263904505214, −4.34630948365906387256034164677, −4.33600643368571720060158511295, −2.97755093220085225562804510752, −1.96675266572149448360600541156, 0, 0,
1.96675266572149448360600541156, 2.97755093220085225562804510752, 4.33600643368571720060158511295, 4.34630948365906387256034164677, 5.58225842583335733263904505214, 6.12829248980041301724103146750, 7.17900529342630246869617479733, 7.58447148639968036921163437354, 8.630678699026020291422832201653, 8.949886996552283651121507945444, 9.734992687492277956512636905242, 9.987057032490220392773635664328, 10.70874564856168831189094202678, 11.48864531792427613986059689074, 12.29232338348535209439525579197, 12.56671209265015014204324459688, 13.24496901716958515000112480750, 13.49228990439729328905237985585
