| L(s) = 1 | + 486·9-s − 5.90e3·29-s − 4.19e4·41-s + 3.36e4·49-s + 3.79e4·61-s + 1.77e5·81-s − 1.02e5·89-s − 1.96e5·101-s − 4.92e5·109-s − 3.22e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 6.83e5·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 2·9-s − 1.30·29-s − 3.89·41-s + 2·49-s + 1.30·61-s + 3·81-s − 1.36·89-s − 1.91·101-s − 3.97·109-s − 2·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s − 1.83·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + 1.83e−6·197-s + 1.79e−6·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.614814603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.614814603\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 244 T + p^{5} T^{2} )( 1 + 244 T + p^{5} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 808 T + p^{5} T^{2} )( 1 + 808 T + p^{5} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2950 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11292 T + p^{5} T^{2} )( 1 + 11292 T + p^{5} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 20950 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 40244 T + p^{5} T^{2} )( 1 + 40244 T + p^{5} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 18950 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 20144 T + p^{5} T^{2} )( 1 + 20144 T + p^{5} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 51050 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 160808 T + p^{5} T^{2} )( 1 + 160808 T + p^{5} T^{2} ) \) |
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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
−9.865183485112014052122778136501, −9.282887394621082613278032723394, −9.052533305039025467714829190533, −8.252411624352257470782930037893, −8.166694900410464455209788023336, −7.45836490995697305659893686496, −7.11706389193744869845964303185, −6.67952714958749669696416758267, −6.59301053333020968344039660404, −5.50010840423586224670129878272, −5.42917153698859629405464819500, −4.82027764880071655049357762733, −4.22288725083518325994894470244, −3.83295438662945671772894708822, −3.52350676840174299477922582326, −2.65278835601502608275604455844, −2.07123365537866385116726341342, −1.44123920693509882309558987390, −1.23391464300200731199668140643, −0.25741620326941965962957600079,
0.25741620326941965962957600079, 1.23391464300200731199668140643, 1.44123920693509882309558987390, 2.07123365537866385116726341342, 2.65278835601502608275604455844, 3.52350676840174299477922582326, 3.83295438662945671772894708822, 4.22288725083518325994894470244, 4.82027764880071655049357762733, 5.42917153698859629405464819500, 5.50010840423586224670129878272, 6.59301053333020968344039660404, 6.67952714958749669696416758267, 7.11706389193744869845964303185, 7.45836490995697305659893686496, 8.166694900410464455209788023336, 8.252411624352257470782930037893, 9.052533305039025467714829190533, 9.282887394621082613278032723394, 9.865183485112014052122778136501
