| L(s) = 1 | + 6·3-s − 484·7-s − 693·9-s + 5.23e3·13-s − 1.15e4·19-s − 2.90e3·21-s + 5.33e3·25-s − 8.53e3·27-s + 4.08e4·31-s − 9.35e4·37-s + 3.14e4·39-s − 1.37e5·43-s − 5.96e4·49-s − 6.94e4·57-s + 4.95e4·61-s + 3.35e5·63-s + 1.68e5·67-s − 2.27e5·73-s + 3.19e4·75-s + 3.19e5·79-s + 4.54e5·81-s − 2.53e6·91-s + 2.45e5·93-s + 1.79e6·97-s + 3.19e6·103-s + 4.43e6·109-s − 5.61e5·111-s + ⋯ |
| L(s) = 1 | + 2/9·3-s − 1.41·7-s − 0.950·9-s + 2.38·13-s − 1.68·19-s − 0.313·21-s + 0.341·25-s − 0.433·27-s + 1.37·31-s − 1.84·37-s + 0.529·39-s − 1.72·43-s − 0.506·49-s − 0.374·57-s + 0.218·61-s + 1.34·63-s + 0.560·67-s − 0.585·73-s + 0.0758·75-s + 0.647·79-s + 0.854·81-s − 3.36·91-s + 0.305·93-s + 1.97·97-s + 2.92·103-s + 3.42·109-s − 0.410·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.413025293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.413025293\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - 2 p T + p^{6} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 1066 p T^{2} + p^{12} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 242 T + p^{6} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 3362 p^{2} T^{2} + p^{12} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2618 T + p^{6} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 1905982 T^{2} + p^{12} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 5786 T + p^{6} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 208876898 T^{2} + p^{12} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1035966962 T^{2} + p^{12} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 20446 T + p^{6} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 46774 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 9487663202 T^{2} + p^{12} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 68618 T + p^{6} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 21106800578 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 14819087378 T^{2} + p^{12} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 61991044562 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 24794 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 84358 T + p^{6} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 151063967522 T^{2} + p^{12} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 113806 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 159742 T + p^{6} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 388294032818 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 583819025758 T^{2} + p^{12} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 899522 T + p^{6} T^{2} )^{2} \) |
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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
−14.65266381376771092999303965711, −14.02549310874133171055871603873, −13.42393757641612420097710587402, −13.12886861110834786905501853432, −12.53103406375134193489638390999, −11.69631460272486440108337641524, −11.17304685344180160947405309743, −10.47761928837802605962565675153, −10.02082547999777236166588642992, −9.040126814414842303447432392661, −8.527421664979238672122677835970, −8.300158916000469484923674920957, −6.91443005779485860572243554041, −6.21302865394191111538034271369, −6.08990079236854987198885290601, −4.76899565129540580136146036217, −3.50736264815613614278849101604, −3.29622682910930402621282178445, −1.92859426622313922842626802216, −0.52419967384380429525887302467,
0.52419967384380429525887302467, 1.92859426622313922842626802216, 3.29622682910930402621282178445, 3.50736264815613614278849101604, 4.76899565129540580136146036217, 6.08990079236854987198885290601, 6.21302865394191111538034271369, 6.91443005779485860572243554041, 8.300158916000469484923674920957, 8.527421664979238672122677835970, 9.040126814414842303447432392661, 10.02082547999777236166588642992, 10.47761928837802605962565675153, 11.17304685344180160947405309743, 11.69631460272486440108337641524, 12.53103406375134193489638390999, 13.12886861110834786905501853432, 13.42393757641612420097710587402, 14.02549310874133171055871603873, 14.65266381376771092999303965711
