| L(s) = 1 | + 3-s − 1.98·5-s − 4.60·7-s + 9-s − 2.92·11-s − 1.83·13-s − 1.98·15-s − 6.93·17-s − 3.32·19-s − 4.60·21-s + 3.86·23-s − 1.06·25-s + 27-s − 6.47·29-s − 4.20·31-s − 2.92·33-s + 9.13·35-s + 0.477·37-s − 1.83·39-s + 1.04·41-s − 1.51·43-s − 1.98·45-s + 9.41·47-s + 14.2·49-s − 6.93·51-s + 3.17·53-s + 5.80·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.887·5-s − 1.74·7-s + 0.333·9-s − 0.881·11-s − 0.508·13-s − 0.512·15-s − 1.68·17-s − 0.761·19-s − 1.00·21-s + 0.805·23-s − 0.212·25-s + 0.192·27-s − 1.20·29-s − 0.754·31-s − 0.508·33-s + 1.54·35-s + 0.0785·37-s − 0.293·39-s + 0.163·41-s − 0.230·43-s − 0.295·45-s + 1.37·47-s + 2.02·49-s − 0.970·51-s + 0.436·53-s + 0.782·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4609446583\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4609446583\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 251 | \( 1 + T \) |
| good | 5 | \( 1 + 1.98T + 5T^{2} \) |
| 7 | \( 1 + 4.60T + 7T^{2} \) |
| 11 | \( 1 + 2.92T + 11T^{2} \) |
| 13 | \( 1 + 1.83T + 13T^{2} \) |
| 17 | \( 1 + 6.93T + 17T^{2} \) |
| 19 | \( 1 + 3.32T + 19T^{2} \) |
| 23 | \( 1 - 3.86T + 23T^{2} \) |
| 29 | \( 1 + 6.47T + 29T^{2} \) |
| 31 | \( 1 + 4.20T + 31T^{2} \) |
| 37 | \( 1 - 0.477T + 37T^{2} \) |
| 41 | \( 1 - 1.04T + 41T^{2} \) |
| 43 | \( 1 + 1.51T + 43T^{2} \) |
| 47 | \( 1 - 9.41T + 47T^{2} \) |
| 53 | \( 1 - 3.17T + 53T^{2} \) |
| 59 | \( 1 + 3.55T + 59T^{2} \) |
| 61 | \( 1 - 5.63T + 61T^{2} \) |
| 67 | \( 1 + 0.844T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 + 4.83T + 79T^{2} \) |
| 83 | \( 1 + 9.79T + 83T^{2} \) |
| 89 | \( 1 + 3.56T + 89T^{2} \) |
| 97 | \( 1 + 3.94T + 97T^{2} \) |
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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
−8.074819006589844271392180411839, −7.23301340780507013258442668112, −6.92738569241310507567101539686, −6.07160533126178407886375522317, −5.15522753880547834626779130601, −4.12986488784195191489816296718, −3.70756096167865603295078635722, −2.76676507427112313128074164272, −2.21078643433108730511343605622, −0.31673859618523257009722936306,
0.31673859618523257009722936306, 2.21078643433108730511343605622, 2.76676507427112313128074164272, 3.70756096167865603295078635722, 4.12986488784195191489816296718, 5.15522753880547834626779130601, 6.07160533126178407886375522317, 6.92738569241310507567101539686, 7.23301340780507013258442668112, 8.074819006589844271392180411839
