| L(s) = 1 | + 2·2-s + 2·4-s − 2·5-s − 4·10-s + 10·13-s − 4·16-s + 6·17-s − 4·20-s − 25-s + 20·26-s − 8·32-s + 12·34-s + 10·37-s − 20·41-s − 2·50-s + 20·52-s + 18·53-s − 24·61-s − 8·64-s − 20·65-s + 12·68-s − 10·73-s + 20·74-s + 8·80-s − 40·82-s − 12·85-s − 10·97-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.894·5-s − 1.26·10-s + 2.77·13-s − 16-s + 1.45·17-s − 0.894·20-s − 1/5·25-s + 3.92·26-s − 1.41·32-s + 2.05·34-s + 1.64·37-s − 3.12·41-s − 0.282·50-s + 2.77·52-s + 2.47·53-s − 3.07·61-s − 64-s − 2.48·65-s + 1.45·68-s − 1.17·73-s + 2.32·74-s + 0.894·80-s − 4.41·82-s − 1.30·85-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.374249228\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.374249228\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−13.16047999006397663681282872416, −12.28027943639995197037816891852, −12.04316356748259253012067595730, −11.64321854713640064014423908993, −11.12592721398720404943772405219, −10.72462583286486581610546272138, −10.10198157549735272147512258998, −9.376833417237356158778943578610, −8.663247039267933374849409520880, −8.337056271821739998998959790353, −7.80049789354900015256841351104, −7.01642705011643884916732879043, −6.46953689600195161302849812082, −5.68521253712848275746432641839, −5.67686025332234930635768142191, −4.55360371376143708591131710061, −4.00594085505945662383360458095, −3.48825090447479168940569400022, −3.03351974073920739973609272382, −1.43392854484617219200441845323,
1.43392854484617219200441845323, 3.03351974073920739973609272382, 3.48825090447479168940569400022, 4.00594085505945662383360458095, 4.55360371376143708591131710061, 5.67686025332234930635768142191, 5.68521253712848275746432641839, 6.46953689600195161302849812082, 7.01642705011643884916732879043, 7.80049789354900015256841351104, 8.337056271821739998998959790353, 8.663247039267933374849409520880, 9.376833417237356158778943578610, 10.10198157549735272147512258998, 10.72462583286486581610546272138, 11.12592721398720404943772405219, 11.64321854713640064014423908993, 12.04316356748259253012067595730, 12.28027943639995197037816891852, 13.16047999006397663681282872416
