| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 2.32·7-s − 8-s + 9-s − 10-s − 5.34·11-s + 12-s + 2.32·14-s + 15-s + 16-s + 4·17-s − 18-s + 4.02·19-s + 20-s − 2.32·21-s + 5.34·22-s − 4.93·23-s − 24-s + 25-s + 27-s − 2.32·28-s + 4.29·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 0.877·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 1.61·11-s + 0.288·12-s + 0.620·14-s + 0.258·15-s + 0.250·16-s + 0.970·17-s − 0.235·18-s + 0.922·19-s + 0.223·20-s − 0.506·21-s + 1.13·22-s − 1.02·23-s − 0.204·24-s + 0.200·25-s + 0.192·27-s − 0.438·28-s + 0.796·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.435515391\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.435515391\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + 2.32T + 7T^{2} \) |
| 11 | \( 1 + 5.34T + 11T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 4.02T + 19T^{2} \) |
| 23 | \( 1 + 4.93T + 23T^{2} \) |
| 29 | \( 1 - 4.29T + 29T^{2} \) |
| 31 | \( 1 + 3.47T + 31T^{2} \) |
| 37 | \( 1 - 3.14T + 37T^{2} \) |
| 41 | \( 1 - 2.64T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 + 1.81T + 47T^{2} \) |
| 53 | \( 1 + 5.48T + 53T^{2} \) |
| 59 | \( 1 - 6.78T + 59T^{2} \) |
| 61 | \( 1 - 0.535T + 61T^{2} \) |
| 67 | \( 1 - 4.10T + 67T^{2} \) |
| 71 | \( 1 + 15.8T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 + 7.96T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 - 1.73T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−8.108173813203218789141526477278, −7.68652740971404147889208487382, −7.07017095533010779359126805737, −5.99201914243998547241336205304, −5.61858980325906754675658191968, −4.52380280267014941555016734959, −3.29872521028188318056054497034, −2.83985346275550562577635992449, −1.96546230718536684133851008700, −0.68609541638473039354441200815,
0.68609541638473039354441200815, 1.96546230718536684133851008700, 2.83985346275550562577635992449, 3.29872521028188318056054497034, 4.52380280267014941555016734959, 5.61858980325906754675658191968, 5.99201914243998547241336205304, 7.07017095533010779359126805737, 7.68652740971404147889208487382, 8.108173813203218789141526477278
