| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 3·8-s + 9-s + 4·11-s + 12-s + 2·13-s − 16-s − 2·17-s + 18-s + 4·22-s + 3·24-s + 2·26-s − 27-s − 6·29-s − 4·31-s + 5·32-s − 4·33-s − 2·34-s − 36-s + 10·37-s − 2·39-s − 2·41-s − 8·43-s − 4·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s + 0.554·13-s − 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.852·22-s + 0.612·24-s + 0.392·26-s − 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.883·32-s − 0.696·33-s − 0.342·34-s − 1/6·36-s + 1.64·37-s − 0.320·39-s − 0.312·41-s − 1.21·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.721732932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.721732932\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.666254884236490206796533986146, −7.82636070975790253834370209277, −6.70042957969876167626964671865, −6.29543404476500847326290274031, −5.47218115114376848776511132537, −4.77491597795748252530443752776, −3.92695287009929121541665624033, −3.46682484986390700294331359273, −2.00815934208801033230858856233, −0.72693122749192687135083507394,
0.72693122749192687135083507394, 2.00815934208801033230858856233, 3.46682484986390700294331359273, 3.92695287009929121541665624033, 4.77491597795748252530443752776, 5.47218115114376848776511132537, 6.29543404476500847326290274031, 6.70042957969876167626964671865, 7.82636070975790253834370209277, 8.666254884236490206796533986146
