| L(s) = 1 | − 8·7-s + 4·9-s − 8·17-s − 8·23-s + 2·25-s + 16·31-s − 4·41-s + 34·49-s − 32·63-s + 8·71-s + 8·73-s − 16·79-s + 7·81-s − 24·89-s − 8·97-s − 24·103-s + 28·113-s + 64·119-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 32·153-s + 157-s + ⋯ |
| L(s) = 1 | − 3.02·7-s + 4/3·9-s − 1.94·17-s − 1.66·23-s + 2/5·25-s + 2.87·31-s − 0.624·41-s + 34/7·49-s − 4.03·63-s + 0.949·71-s + 0.936·73-s − 1.80·79-s + 7/9·81-s − 2.54·89-s − 0.812·97-s − 2.36·103-s + 2.63·113-s + 5.86·119-s + 1.81·121-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 − 2.58·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8124022989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8124022989\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 100 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 116 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 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
−11.06667657864456293587325431395, −10.39017818164891154948924488157, −10.10902537489330026071095475360, −9.739293793711422052090321075488, −9.644120037497726486199151912738, −9.074550109121390332568244153649, −8.289653010754335617108701445613, −8.275633765158505299243361421439, −7.09621972605103215146707564603, −6.89615184515577686861177937745, −6.71202377626927932295426514956, −6.06050828383155309270346852197, −5.92858865262993170321113365516, −4.80370649194988237267841040793, −4.17305747562492982760510545305, −4.04997260810033621865495591668, −3.10550810409748815510461585071, −2.78191032351616055652616608942, −1.92373065670686808966821961592, −0.51977093546814629137452395290,
0.51977093546814629137452395290, 1.92373065670686808966821961592, 2.78191032351616055652616608942, 3.10550810409748815510461585071, 4.04997260810033621865495591668, 4.17305747562492982760510545305, 4.80370649194988237267841040793, 5.92858865262993170321113365516, 6.06050828383155309270346852197, 6.71202377626927932295426514956, 6.89615184515577686861177937745, 7.09621972605103215146707564603, 8.275633765158505299243361421439, 8.289653010754335617108701445613, 9.074550109121390332568244153649, 9.644120037497726486199151912738, 9.739293793711422052090321075488, 10.10902537489330026071095475360, 10.39017818164891154948924488157, 11.06667657864456293587325431395
