| L(s) = 1 | + 3·3-s − 7·7-s + 9·9-s + 54·11-s + 55·13-s − 18·17-s + 25·19-s − 21·21-s − 18·23-s + 27·27-s − 54·29-s + 271·31-s + 162·33-s − 314·37-s + 165·39-s − 360·41-s − 163·43-s + 522·47-s − 294·49-s − 54·51-s + 36·53-s + 75·57-s − 126·59-s + 47·61-s − 63·63-s − 343·67-s − 54·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.48·11-s + 1.17·13-s − 0.256·17-s + 0.301·19-s − 0.218·21-s − 0.163·23-s + 0.192·27-s − 0.345·29-s + 1.57·31-s + 0.854·33-s − 1.39·37-s + 0.677·39-s − 1.37·41-s − 0.578·43-s + 1.62·47-s − 6/7·49-s − 0.148·51-s + 0.0933·53-s + 0.174·57-s − 0.278·59-s + 0.0986·61-s − 0.125·63-s − 0.625·67-s − 0.0942·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.064269320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.064269320\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + p T + p^{3} T^{2} \) |
| 11 | \( 1 - 54 T + p^{3} T^{2} \) |
| 13 | \( 1 - 55 T + p^{3} T^{2} \) |
| 17 | \( 1 + 18 T + p^{3} T^{2} \) |
| 19 | \( 1 - 25 T + p^{3} T^{2} \) |
| 23 | \( 1 + 18 T + p^{3} T^{2} \) |
| 29 | \( 1 + 54 T + p^{3} T^{2} \) |
| 31 | \( 1 - 271 T + p^{3} T^{2} \) |
| 37 | \( 1 + 314 T + p^{3} T^{2} \) |
| 41 | \( 1 + 360 T + p^{3} T^{2} \) |
| 43 | \( 1 + 163 T + p^{3} T^{2} \) |
| 47 | \( 1 - 522 T + p^{3} T^{2} \) |
| 53 | \( 1 - 36 T + p^{3} T^{2} \) |
| 59 | \( 1 + 126 T + p^{3} T^{2} \) |
| 61 | \( 1 - 47 T + p^{3} T^{2} \) |
| 67 | \( 1 + 343 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1080 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1054 T + p^{3} T^{2} \) |
| 79 | \( 1 - 568 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1422 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1440 T + p^{3} T^{2} \) |
| 97 | \( 1 - 439 T + p^{3} T^{2} \) |
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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
−9.214190514400189629346550162147, −8.699083447069005947824717214905, −7.87095718732532539811509170004, −6.67056311055935811850771249822, −6.36190828019460173744454721414, −5.03157821909487529863666994395, −3.86216453075528095025874712350, −3.37389685055367477151411391045, −1.96510632014877803206864906231, −0.920986609615648720056310955458,
0.920986609615648720056310955458, 1.96510632014877803206864906231, 3.37389685055367477151411391045, 3.86216453075528095025874712350, 5.03157821909487529863666994395, 6.36190828019460173744454721414, 6.67056311055935811850771249822, 7.87095718732532539811509170004, 8.699083447069005947824717214905, 9.214190514400189629346550162147
