| L(s) = 1 | + 9-s − 2·17-s − 8·19-s − 2·25-s − 4·41-s − 8·43-s − 2·49-s − 8·59-s + 8·67-s + 4·73-s + 81-s − 24·83-s + 12·89-s + 4·97-s + 16·107-s + 12·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 2·153-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1/3·9-s − 0.485·17-s − 1.83·19-s − 2/5·25-s − 0.624·41-s − 1.21·43-s − 2/7·49-s − 1.04·59-s + 0.977·67-s + 0.468·73-s + 1/9·81-s − 2.63·83-s + 1.27·89-s + 0.406·97-s + 1.54·107-s + 1.12·113-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.161·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.253394829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.253394829\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−7.66643847783662331491727887735, −7.06694925286556935352027326222, −6.77600645740645616349403873756, −6.44235013404215990367300044482, −5.97933664252796158852075932824, −5.56669173386029527603454231932, −4.95891298489770699391384730184, −4.51072908441005746778926140583, −4.29069031972477030392473592384, −3.60512740650267412739964674531, −3.24555909857976407870127946051, −2.48784527810785707123696698847, −1.98833378335270627179565432086, −1.53318243450965361161045557035, −0.42052544310490231543126167757,
0.42052544310490231543126167757, 1.53318243450965361161045557035, 1.98833378335270627179565432086, 2.48784527810785707123696698847, 3.24555909857976407870127946051, 3.60512740650267412739964674531, 4.29069031972477030392473592384, 4.51072908441005746778926140583, 4.95891298489770699391384730184, 5.56669173386029527603454231932, 5.97933664252796158852075932824, 6.44235013404215990367300044482, 6.77600645740645616349403873756, 7.06694925286556935352027326222, 7.66643847783662331491727887735
