| L(s) = 1 | + 0.735·2-s + 3-s − 1.45·4-s + 4.21·5-s + 0.735·6-s − 7-s − 2.54·8-s + 9-s + 3.09·10-s − 5.24·11-s − 1.45·12-s + 2.61·13-s − 0.735·14-s + 4.21·15-s + 1.04·16-s − 3.16·17-s + 0.735·18-s + 2.37·19-s − 6.15·20-s − 21-s − 3.85·22-s − 4.69·23-s − 2.54·24-s + 12.7·25-s + 1.92·26-s + 27-s + 1.45·28-s + ⋯ |
| L(s) = 1 | + 0.519·2-s + 0.577·3-s − 0.729·4-s + 1.88·5-s + 0.300·6-s − 0.377·7-s − 0.899·8-s + 0.333·9-s + 0.979·10-s − 1.58·11-s − 0.421·12-s + 0.725·13-s − 0.196·14-s + 1.08·15-s + 0.262·16-s − 0.768·17-s + 0.173·18-s + 0.545·19-s − 1.37·20-s − 0.218·21-s − 0.821·22-s − 0.979·23-s − 0.519·24-s + 2.55·25-s + 0.377·26-s + 0.192·27-s + 0.275·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.460649471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.460649471\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 - 0.735T + 2T^{2} \) |
| 5 | \( 1 - 4.21T + 5T^{2} \) |
| 11 | \( 1 + 5.24T + 11T^{2} \) |
| 13 | \( 1 - 2.61T + 13T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 19 | \( 1 - 2.37T + 19T^{2} \) |
| 23 | \( 1 + 4.69T + 23T^{2} \) |
| 29 | \( 1 - 3.00T + 29T^{2} \) |
| 31 | \( 1 - 9.09T + 31T^{2} \) |
| 37 | \( 1 + 7.46T + 37T^{2} \) |
| 41 | \( 1 - 2.31T + 41T^{2} \) |
| 43 | \( 1 + 2.29T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 - 4.87T + 53T^{2} \) |
| 59 | \( 1 - 1.60T + 59T^{2} \) |
| 61 | \( 1 - 3.92T + 61T^{2} \) |
| 67 | \( 1 + 2.10T + 67T^{2} \) |
| 71 | \( 1 + 3.73T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 + 8.70T + 79T^{2} \) |
| 83 | \( 1 - 3.36T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 + 1.01T + 97T^{2} \) |
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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
−8.033263494011081468740982779272, −6.95570234577773354826217504062, −6.20086729718751789691715520704, −5.70504756968095896746117336198, −5.10145404980429105486832328194, −4.41576104815077469110905751539, −3.37943549583852542083960293652, −2.65595361766147660489300737674, −2.10569886523174717055380816899, −0.823704220911497279582476788148,
0.823704220911497279582476788148, 2.10569886523174717055380816899, 2.65595361766147660489300737674, 3.37943549583852542083960293652, 4.41576104815077469110905751539, 5.10145404980429105486832328194, 5.70504756968095896746117336198, 6.20086729718751789691715520704, 6.95570234577773354826217504062, 8.033263494011081468740982779272
