| L(s) = 1 | + 3-s − 3·4-s − 2·7-s + 9-s + 8·11-s − 3·12-s + 5·16-s − 12·17-s + 8·19-s − 2·21-s − 6·25-s + 27-s + 6·28-s − 4·29-s + 8·33-s − 3·36-s − 24·44-s + 5·48-s + 3·49-s − 12·51-s + 12·53-s + 8·57-s − 2·63-s − 3·64-s + 8·67-s + 36·68-s − 6·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 3/2·4-s − 0.755·7-s + 1/3·9-s + 2.41·11-s − 0.866·12-s + 5/4·16-s − 2.91·17-s + 1.83·19-s − 0.436·21-s − 6/5·25-s + 0.192·27-s + 1.13·28-s − 0.742·29-s + 1.39·33-s − 1/2·36-s − 3.61·44-s + 0.721·48-s + 3/7·49-s − 1.68·51-s + 1.64·53-s + 1.05·57-s − 0.251·63-s − 3/8·64-s + 0.977·67-s + 4.36·68-s − 0.692·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1271403 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1271403 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 31 | $C_2$ | \( 1 + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−7.80453780620392921373441407832, −7.25047783802838427330028450541, −6.97568825543215712468784158163, −6.48994701511899578450675140585, −6.12577327356424149496817271030, −5.49834728573181346503049108711, −5.04547784492208655878128457795, −4.25189867464194734459254279253, −4.13559084050773741974089362056, −3.85385905737148835924663140068, −3.27936376543986290075856446876, −2.51192082021878746849812004835, −1.78287040315203100689076699444, −1.02448108936498933795251336968, 0,
1.02448108936498933795251336968, 1.78287040315203100689076699444, 2.51192082021878746849812004835, 3.27936376543986290075856446876, 3.85385905737148835924663140068, 4.13559084050773741974089362056, 4.25189867464194734459254279253, 5.04547784492208655878128457795, 5.49834728573181346503049108711, 6.12577327356424149496817271030, 6.48994701511899578450675140585, 6.97568825543215712468784158163, 7.25047783802838427330028450541, 7.80453780620392921373441407832
