| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 5·7-s + 8-s + 9-s − 11-s + 12-s − 5·14-s + 16-s + 4·17-s + 18-s + 3·19-s − 5·21-s − 22-s + 23-s + 24-s + 27-s − 5·28-s + 6·29-s − 31-s + 32-s − 33-s + 4·34-s + 36-s + 4·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.88·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s − 1.33·14-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 0.688·19-s − 1.09·21-s − 0.213·22-s + 0.208·23-s + 0.204·24-s + 0.192·27-s − 0.944·28-s + 1.11·29-s − 0.179·31-s + 0.176·32-s − 0.174·33-s + 0.685·34-s + 1/6·36-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.134478156\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.134478156\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.161945058453112513595584410005, −7.49391269460864216446385515019, −6.74077270636893430153068344267, −6.17721768161906194996196903416, −5.41775919238442297270092743082, −4.48797396004496236766001778692, −3.42468065614581908043689633451, −3.20325720671657470667140663817, −2.33091504521382109584331939466, −0.850626932479884459655088516509,
0.850626932479884459655088516509, 2.33091504521382109584331939466, 3.20325720671657470667140663817, 3.42468065614581908043689633451, 4.48797396004496236766001778692, 5.41775919238442297270092743082, 6.17721768161906194996196903416, 6.74077270636893430153068344267, 7.49391269460864216446385515019, 8.161945058453112513595584410005
