| L(s) = 1 | + 12·5-s − 27·9-s − 172·13-s + 852·17-s − 1.14e3·25-s + 2.36e3·29-s − 860·37-s + 4.50e3·41-s − 324·45-s + 914·49-s − 3.20e3·53-s + 4.22e3·61-s − 2.06e3·65-s + 8.13e3·73-s + 729·81-s + 1.02e4·85-s − 4.09e3·89-s − 5.88e3·97-s − 2.33e4·101-s + 3.50e4·109-s + 2.40e4·113-s + 4.64e3·117-s − 5.71e3·121-s − 2.16e4·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.479·5-s − 1/3·9-s − 1.01·13-s + 2.94·17-s − 1.82·25-s + 2.81·29-s − 0.628·37-s + 2.67·41-s − 0.159·45-s + 0.380·49-s − 1.14·53-s + 1.13·61-s − 0.488·65-s + 1.52·73-s + 1/9·81-s + 1.41·85-s − 0.516·89-s − 0.625·97-s − 2.29·101-s + 2.95·109-s + 1.88·113-s + 0.339·117-s − 0.390·121-s − 1.38·125-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.079747824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.079747824\) |
| \(L(3)\) |
|
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 | $C_2$ | \( 1 + p^{3} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 6 T + p^{4} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 914 T^{2} + p^{8} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5710 T^{2} + p^{8} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 86 T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 426 T + p^{4} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 256754 T^{2} + p^{8} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 190 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 1182 T + p^{4} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 582958 T^{2} + p^{8} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 430 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2250 T + p^{4} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 190 p^{2} T^{2} + p^{8} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 9619394 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 1602 T + p^{4} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 11602610 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2114 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 38587634 T^{2} + p^{8} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 41865410 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 4066 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 47103314 T^{2} + p^{8} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 10900850 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2046 T + p^{4} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2942 T + p^{4} T^{2} )^{2} \) |
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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
−15.08648996790147396481997118942, −14.44044932249046000372628729897, −13.99674766529058251096525996969, −13.86924617555421987527962627285, −12.59975500553505287172079269396, −12.35400131475003144023861275541, −11.89890521089086699520929490451, −11.08410735332800373285155252329, −10.14836976428060325075866909605, −9.911278195966264303669118365165, −9.390896301329452375271742999677, −8.226501656210905022532656375581, −7.87186580605940306520826360810, −7.09138556043584818311730088930, −6.00217194436756973025855971757, −5.57794918279561849813948839302, −4.63887968615779976205185011438, −3.42841110485394955965120713675, −2.46075016701042556698623068330, −0.962472847177862525655787463656,
0.962472847177862525655787463656, 2.46075016701042556698623068330, 3.42841110485394955965120713675, 4.63887968615779976205185011438, 5.57794918279561849813948839302, 6.00217194436756973025855971757, 7.09138556043584818311730088930, 7.87186580605940306520826360810, 8.226501656210905022532656375581, 9.390896301329452375271742999677, 9.911278195966264303669118365165, 10.14836976428060325075866909605, 11.08410735332800373285155252329, 11.89890521089086699520929490451, 12.35400131475003144023861275541, 12.59975500553505287172079269396, 13.86924617555421987527962627285, 13.99674766529058251096525996969, 14.44044932249046000372628729897, 15.08648996790147396481997118942
