| L(s) = 1 | − 3-s − 3·7-s + 9-s + 2·11-s + 2·13-s + 17-s − 6·19-s + 3·21-s + 23-s − 27-s − 3·29-s − 3·31-s − 2·33-s + 3·37-s − 2·39-s − 7·41-s + 2·49-s − 51-s − 5·53-s + 6·57-s − 9·59-s − 12·61-s − 3·63-s + 3·67-s − 69-s + 3·71-s − 2·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s + 0.242·17-s − 1.37·19-s + 0.654·21-s + 0.208·23-s − 0.192·27-s − 0.557·29-s − 0.538·31-s − 0.348·33-s + 0.493·37-s − 0.320·39-s − 1.09·41-s + 2/7·49-s − 0.140·51-s − 0.686·53-s + 0.794·57-s − 1.17·59-s − 1.53·61-s − 0.377·63-s + 0.366·67-s − 0.120·69-s + 0.356·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4893657674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4893657674\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.57884438657164, −13.02782547188098, −12.67876879112310, −12.31386476513735, −11.71806949439909, −11.14267801646856, −10.79723231961337, −10.26392135240461, −9.752009270914493, −9.212670575339535, −8.897370988497919, −8.191676907760036, −7.632543182379252, −6.978953378033304, −6.431849556072146, −6.261810753571358, −5.684572415562991, −4.984610937192883, −4.385737255684178, −3.782845775134804, −3.367983062938425, −2.660041370330356, −1.811646002643871, −1.244222561375002, −0.2316041553735973,
0.2316041553735973, 1.244222561375002, 1.811646002643871, 2.660041370330356, 3.367983062938425, 3.782845775134804, 4.385737255684178, 4.984610937192883, 5.684572415562991, 6.261810753571358, 6.431849556072146, 6.978953378033304, 7.632543182379252, 8.191676907760036, 8.897370988497919, 9.212670575339535, 9.752009270914493, 10.26392135240461, 10.79723231961337, 11.14267801646856, 11.71806949439909, 12.31386476513735, 12.67876879112310, 13.02782547188098, 13.57884438657164
