| L(s) = 1 | − 2·3-s − 2·5-s + 3·7-s + 3·9-s − 3·11-s − 2·13-s + 4·15-s − 5·17-s + 4·19-s − 6·21-s − 23-s + 3·25-s − 4·27-s + 6·29-s − 2·31-s + 6·33-s − 6·35-s + 3·37-s + 4·39-s + 5·41-s + 2·43-s − 6·45-s + 6·47-s + 49-s + 10·51-s − 21·53-s + 6·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 1.13·7-s + 9-s − 0.904·11-s − 0.554·13-s + 1.03·15-s − 1.21·17-s + 0.917·19-s − 1.30·21-s − 0.208·23-s + 3/5·25-s − 0.769·27-s + 1.11·29-s − 0.359·31-s + 1.04·33-s − 1.01·35-s + 0.493·37-s + 0.640·39-s + 0.780·41-s + 0.304·43-s − 0.894·45-s + 0.875·47-s + 1/7·49-s + 1.40·51-s − 2.88·53-s + 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38937600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38937600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 32 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + T + 38 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 3 T + 68 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 5 T + 80 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 54 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 21 T + 208 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + T + 48 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 14 T + 150 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3 T + 136 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 25 T + 306 T^{2} - 25 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 18 T + 214 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 9 T + 124 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - T + 120 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−7.68284278135086176214426425147, −7.64443311269993627605978808852, −7.29145826787706960038538148229, −6.77350136833621275761050538218, −6.42841679022934265409983279076, −6.17343646716811657835812101606, −5.65507880697129072992765303853, −5.20060598624385051487804315066, −5.07474995102635307375994791308, −4.58642848686031278148156555229, −4.29427374354563390957305307722, −4.25736167305033659132775398790, −3.22013683807680589107235799433, −3.21228177809019626860899250176, −2.37638162251218287752141963510, −2.18057610830955619560329211247, −1.28835582103390192415685296508, −1.10546151124067946891137239547, 0, 0,
1.10546151124067946891137239547, 1.28835582103390192415685296508, 2.18057610830955619560329211247, 2.37638162251218287752141963510, 3.21228177809019626860899250176, 3.22013683807680589107235799433, 4.25736167305033659132775398790, 4.29427374354563390957305307722, 4.58642848686031278148156555229, 5.07474995102635307375994791308, 5.20060598624385051487804315066, 5.65507880697129072992765303853, 6.17343646716811657835812101606, 6.42841679022934265409983279076, 6.77350136833621275761050538218, 7.29145826787706960038538148229, 7.64443311269993627605978808852, 7.68284278135086176214426425147
