| L(s) = 1 | + 3-s + 4-s − 7-s + 9-s + 12-s − 13-s + 16-s − 21-s + 25-s + 27-s − 28-s − 31-s + 36-s − 37-s − 39-s − 43-s + 48-s − 52-s − 61-s − 63-s + 64-s − 67-s − 73-s + 75-s − 79-s + 81-s − 84-s + ⋯ |
| L(s) = 1 | + 3-s + 4-s − 7-s + 9-s + 12-s − 13-s + 16-s − 21-s + 25-s + 27-s − 28-s − 31-s + 36-s − 37-s − 39-s − 43-s + 48-s − 52-s − 61-s − 63-s + 64-s − 67-s − 73-s + 75-s − 79-s + 81-s − 84-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.593231853\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.593231853\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.06290955317929830767358687855, −9.288797674257555469372289811718, −8.452967929121358153835405476936, −7.35157812485478151889927165723, −7.02532427698434232535483449218, −6.05528727277577329109096119249, −4.80679393526692214256010234380, −3.45799166917935983993729646004, −2.86640946537598527360617691881, −1.79957236321509578091517513370,
1.79957236321509578091517513370, 2.86640946537598527360617691881, 3.45799166917935983993729646004, 4.80679393526692214256010234380, 6.05528727277577329109096119249, 7.02532427698434232535483449218, 7.35157812485478151889927165723, 8.452967929121358153835405476936, 9.288797674257555469372289811718, 10.06290955317929830767358687855
