| L(s) = 1 | − 2-s + 4-s + 8·5-s − 8-s − 8·10-s − 2·13-s + 16-s − 6·17-s + 8·20-s + 38·25-s + 2·26-s + 10·29-s − 32-s + 6·34-s − 4·37-s − 8·40-s + 16·41-s − 5·49-s − 38·50-s − 2·52-s + 2·53-s − 10·58-s + 4·61-s + 64-s − 16·65-s − 6·68-s + 18·73-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 3.57·5-s − 0.353·8-s − 2.52·10-s − 0.554·13-s + 1/4·16-s − 1.45·17-s + 1.78·20-s + 38/5·25-s + 0.392·26-s + 1.85·29-s − 0.176·32-s + 1.02·34-s − 0.657·37-s − 1.26·40-s + 2.49·41-s − 5/7·49-s − 5.37·50-s − 0.277·52-s + 0.274·53-s − 1.31·58-s + 0.512·61-s + 1/8·64-s − 1.98·65-s − 0.727·68-s + 2.10·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 935712 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 935712 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.216853208\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.216853208\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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
−8.248270166406525333712391337291, −7.892225159593170281658461026008, −6.87482507907449229860442380234, −6.73926084905849932341893089023, −6.55862357806514374286079019861, −5.88794573838587439349205533248, −5.70493789281341106083458944039, −5.09266829498719992624263414553, −4.80751556264464963514931464778, −4.05875927362791341229655398896, −2.85985483208439285249907021396, −2.55734550860524537659292391699, −2.24529612425880911912836760045, −1.62231881828892009952147969605, −0.985714903183510332642793788099,
0.985714903183510332642793788099, 1.62231881828892009952147969605, 2.24529612425880911912836760045, 2.55734550860524537659292391699, 2.85985483208439285249907021396, 4.05875927362791341229655398896, 4.80751556264464963514931464778, 5.09266829498719992624263414553, 5.70493789281341106083458944039, 5.88794573838587439349205533248, 6.55862357806514374286079019861, 6.73926084905849932341893089023, 6.87482507907449229860442380234, 7.892225159593170281658461026008, 8.248270166406525333712391337291
