| L(s) = 1 | + 10·3-s + 18·7-s + 73·9-s + 16·11-s − 6·13-s + 6·17-s + 124·19-s + 180·21-s − 42·23-s + 460·27-s − 142·29-s − 188·31-s + 160·33-s + 202·37-s − 60·39-s + 54·41-s + 66·43-s − 38·47-s − 19·49-s + 60·51-s + 738·53-s + 1.24e3·57-s − 564·59-s + 262·61-s + 1.31e3·63-s − 554·67-s − 420·69-s + ⋯ |
| L(s) = 1 | + 1.92·3-s + 0.971·7-s + 2.70·9-s + 0.438·11-s − 0.128·13-s + 0.0856·17-s + 1.49·19-s + 1.87·21-s − 0.380·23-s + 3.27·27-s − 0.909·29-s − 1.08·31-s + 0.844·33-s + 0.897·37-s − 0.246·39-s + 0.205·41-s + 0.234·43-s − 0.117·47-s − 0.0553·49-s + 0.164·51-s + 1.91·53-s + 2.88·57-s − 1.24·59-s + 0.549·61-s + 2.62·63-s − 1.01·67-s − 0.732·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.025801753\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.025801753\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 10 T + p^{3} T^{2} \) |
| 7 | \( 1 - 18 T + p^{3} T^{2} \) |
| 11 | \( 1 - 16 T + p^{3} T^{2} \) |
| 13 | \( 1 + 6 T + p^{3} T^{2} \) |
| 17 | \( 1 - 6 T + p^{3} T^{2} \) |
| 19 | \( 1 - 124 T + p^{3} T^{2} \) |
| 23 | \( 1 + 42 T + p^{3} T^{2} \) |
| 29 | \( 1 + 142 T + p^{3} T^{2} \) |
| 31 | \( 1 + 188 T + p^{3} T^{2} \) |
| 37 | \( 1 - 202 T + p^{3} T^{2} \) |
| 41 | \( 1 - 54 T + p^{3} T^{2} \) |
| 43 | \( 1 - 66 T + p^{3} T^{2} \) |
| 47 | \( 1 + 38 T + p^{3} T^{2} \) |
| 53 | \( 1 - 738 T + p^{3} T^{2} \) |
| 59 | \( 1 + 564 T + p^{3} T^{2} \) |
| 61 | \( 1 - 262 T + p^{3} T^{2} \) |
| 67 | \( 1 + 554 T + p^{3} T^{2} \) |
| 71 | \( 1 - 140 T + p^{3} T^{2} \) |
| 73 | \( 1 + 882 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1160 T + p^{3} T^{2} \) |
| 83 | \( 1 - 642 T + p^{3} T^{2} \) |
| 89 | \( 1 + 854 T + p^{3} T^{2} \) |
| 97 | \( 1 - 478 T + p^{3} T^{2} \) |
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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
−9.030723675046496000793953685642, −8.266104484489542373852655357687, −7.53560445305109753691816579884, −7.18781611819576656786145528191, −5.70006715547996376501945486637, −4.59022546944363506947839632045, −3.82505640652458558809879893181, −2.98632258337448696459645915275, −1.99032160340125566373398067170, −1.22454801492682224576409444841,
1.22454801492682224576409444841, 1.99032160340125566373398067170, 2.98632258337448696459645915275, 3.82505640652458558809879893181, 4.59022546944363506947839632045, 5.70006715547996376501945486637, 7.18781611819576656786145528191, 7.53560445305109753691816579884, 8.266104484489542373852655357687, 9.030723675046496000793953685642
