| L(s) = 1 | + 2·3-s − 2·5-s + 2·7-s + 3·9-s − 8·11-s + 4·13-s − 4·15-s − 10·17-s + 2·19-s + 4·21-s − 4·25-s + 4·27-s − 10·29-s − 4·31-s − 16·33-s − 4·35-s + 4·37-s + 8·39-s − 4·41-s − 4·43-s − 6·45-s − 10·47-s + 3·49-s − 20·51-s − 2·53-s + 16·55-s + 4·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 0.755·7-s + 9-s − 2.41·11-s + 1.10·13-s − 1.03·15-s − 2.42·17-s + 0.458·19-s + 0.872·21-s − 4/5·25-s + 0.769·27-s − 1.85·29-s − 0.718·31-s − 2.78·33-s − 0.676·35-s + 0.657·37-s + 1.28·39-s − 0.624·41-s − 0.609·43-s − 0.894·45-s − 1.45·47-s + 3/7·49-s − 2.80·51-s − 0.274·53-s + 2.15·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10188864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10188864 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 10 T + 56 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 80 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 10 T + 92 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 32 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2 T + 116 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.233094778261918892145358667709, −8.215080694543461831295586226946, −7.73591949671411004845472382152, −7.69077403006010244619603218209, −7.01509303503386221066223596200, −6.96597955757863350195984661075, −6.10424659975622265851487578233, −5.91836790843344527318839669470, −5.24732949074927538314814056862, −4.99746464758065943525636689581, −4.38183196384062589024908734169, −4.32522413119396594845933725526, −3.51066016760584286453364321721, −3.48395451475226006682558209041, −2.74266292618532147556381359504, −2.38738853561610578284772266510, −1.83437597084645015196398689254, −1.50974378961662278794505562502, 0, 0,
1.50974378961662278794505562502, 1.83437597084645015196398689254, 2.38738853561610578284772266510, 2.74266292618532147556381359504, 3.48395451475226006682558209041, 3.51066016760584286453364321721, 4.32522413119396594845933725526, 4.38183196384062589024908734169, 4.99746464758065943525636689581, 5.24732949074927538314814056862, 5.91836790843344527318839669470, 6.10424659975622265851487578233, 6.96597955757863350195984661075, 7.01509303503386221066223596200, 7.69077403006010244619603218209, 7.73591949671411004845472382152, 8.215080694543461831295586226946, 8.233094778261918892145358667709
