| L(s) = 1 | − 2-s − 2.82·3-s + 4-s + 0.585·5-s + 2.82·6-s − 3.41·7-s − 8-s + 5.00·9-s − 0.585·10-s + 4·11-s − 2.82·12-s + 3.41·14-s − 1.65·15-s + 16-s + 1.41·17-s − 5.00·18-s − 6·19-s + 0.585·20-s + 9.65·21-s − 4·22-s − 5.65·23-s + 2.82·24-s − 4.65·25-s − 5.65·27-s − 3.41·28-s + 9.65·29-s + 1.65·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.63·3-s + 0.5·4-s + 0.261·5-s + 1.15·6-s − 1.29·7-s − 0.353·8-s + 1.66·9-s − 0.185·10-s + 1.20·11-s − 0.816·12-s + 0.912·14-s − 0.427·15-s + 0.250·16-s + 0.342·17-s − 1.17·18-s − 1.37·19-s + 0.130·20-s + 2.10·21-s − 0.852·22-s − 1.17·23-s + 0.577·24-s − 0.931·25-s − 1.08·27-s − 0.645·28-s + 1.79·29-s + 0.302·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4461155536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4461155536\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 - 0.585T + 5T^{2} \) |
| 7 | \( 1 + 3.41T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 - 9.65T + 29T^{2} \) |
| 37 | \( 1 - 1.65T + 37T^{2} \) |
| 41 | \( 1 - 8.82T + 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 5.17T + 53T^{2} \) |
| 59 | \( 1 - 7.17T + 59T^{2} \) |
| 61 | \( 1 - 4.48T + 61T^{2} \) |
| 67 | \( 1 + 6.48T + 67T^{2} \) |
| 71 | \( 1 - 6.24T + 71T^{2} \) |
| 73 | \( 1 + 3.65T + 73T^{2} \) |
| 79 | \( 1 + 4.82T + 79T^{2} \) |
| 83 | \( 1 + 16.7T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
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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
−8.111266969007137768920229800998, −6.97455913411906210154583447851, −6.62362291719492122175110511244, −6.05844581626898164268734798554, −5.66357593552794528464886274751, −4.39761959931704293522981786357, −3.84614812268352908064876691143, −2.58112461273422781519765151843, −1.44828299760589920255174745968, −0.44680072863095070128974502024,
0.44680072863095070128974502024, 1.44828299760589920255174745968, 2.58112461273422781519765151843, 3.84614812268352908064876691143, 4.39761959931704293522981786357, 5.66357593552794528464886274751, 6.05844581626898164268734798554, 6.62362291719492122175110511244, 6.97455913411906210154583447851, 8.111266969007137768920229800998
