| L(s) = 1 | − 4·3-s + 2·5-s + 8·9-s + 8·11-s − 2·13-s − 8·15-s + 8·17-s + 4·19-s + 2·25-s − 12·27-s + 14·29-s − 8·31-s − 32·33-s + 10·37-s + 8·39-s + 16·45-s + 24·47-s − 49-s − 32·51-s + 2·53-s + 16·55-s − 16·57-s − 4·59-s + 6·61-s − 4·65-s + 8·67-s − 8·75-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 0.894·5-s + 8/3·9-s + 2.41·11-s − 0.554·13-s − 2.06·15-s + 1.94·17-s + 0.917·19-s + 2/5·25-s − 2.30·27-s + 2.59·29-s − 1.43·31-s − 5.57·33-s + 1.64·37-s + 1.28·39-s + 2.38·45-s + 3.50·47-s − 1/7·49-s − 4.48·51-s + 0.274·53-s + 2.15·55-s − 2.11·57-s − 0.520·59-s + 0.768·61-s − 0.496·65-s + 0.977·67-s − 0.923·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12845056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12845056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.461682506\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.461682506\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
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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
−9.194509606119028888660756859645, −8.347758496582626402290484925966, −7.77966847402177334007241965603, −7.56097037612622255724033188553, −7.00748194460695364694110625545, −6.75544646560757698209092702263, −6.35011894085368804168283407919, −6.11721737854790440847467979793, −5.67461538678480301626394784947, −5.60388935615225129894258629882, −5.01536643373798400754019077471, −4.82302794409823594469258645799, −4.23380516094162558566534492111, −3.72758912958477954777886808616, −3.51197403400702070159533176027, −2.63182432653956139557969839012, −2.17467428430256450485813345810, −1.29056420892013321177971984612, −0.929654690666695116161220461374, −0.817767215745117044398341816300,
0.817767215745117044398341816300, 0.929654690666695116161220461374, 1.29056420892013321177971984612, 2.17467428430256450485813345810, 2.63182432653956139557969839012, 3.51197403400702070159533176027, 3.72758912958477954777886808616, 4.23380516094162558566534492111, 4.82302794409823594469258645799, 5.01536643373798400754019077471, 5.60388935615225129894258629882, 5.67461538678480301626394784947, 6.11721737854790440847467979793, 6.35011894085368804168283407919, 6.75544646560757698209092702263, 7.00748194460695364694110625545, 7.56097037612622255724033188553, 7.77966847402177334007241965603, 8.347758496582626402290484925966, 9.194509606119028888660756859645
