| L(s) = 1 | + 3-s + 4-s + 2·7-s + 9-s + 12-s + 4·13-s + 16-s + 16·19-s + 2·21-s + 25-s + 27-s + 2·28-s − 8·31-s + 36-s − 20·37-s + 4·39-s − 8·43-s + 48-s + 3·49-s + 4·52-s + 16·57-s − 20·61-s + 2·63-s + 64-s − 8·67-s − 20·73-s + 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s + 0.755·7-s + 1/3·9-s + 0.288·12-s + 1.10·13-s + 1/4·16-s + 3.67·19-s + 0.436·21-s + 1/5·25-s + 0.192·27-s + 0.377·28-s − 1.43·31-s + 1/6·36-s − 3.28·37-s + 0.640·39-s − 1.21·43-s + 0.144·48-s + 3/7·49-s + 0.554·52-s + 2.11·57-s − 2.56·61-s + 0.251·63-s + 1/8·64-s − 0.977·67-s − 2.34·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 132300 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132300 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.747007217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.747007217\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−9.262397060563074490443203035163, −8.790922026040486061843055845413, −8.581528352643131576230344729464, −7.67845508591579519153392227466, −7.48230039465939347417283743898, −7.19801386358340821477645347390, −6.43968643454495280602009146366, −5.76093596813527562032392850894, −5.20272031164140682979668587117, −4.97103303999261607614028493172, −3.88273199773070623460496754838, −3.22977772739188491587716942834, −3.13125557587754062819456448397, −1.69594451483764738863265937021, −1.41006155845401754086894809680,
1.41006155845401754086894809680, 1.69594451483764738863265937021, 3.13125557587754062819456448397, 3.22977772739188491587716942834, 3.88273199773070623460496754838, 4.97103303999261607614028493172, 5.20272031164140682979668587117, 5.76093596813527562032392850894, 6.43968643454495280602009146366, 7.19801386358340821477645347390, 7.48230039465939347417283743898, 7.67845508591579519153392227466, 8.581528352643131576230344729464, 8.790922026040486061843055845413, 9.262397060563074490443203035163
