| L(s) = 1 | + 2-s + 4-s + 4.42·5-s − 3.31·7-s + 8-s + 4.42·10-s + 2.02·11-s + 3.02·13-s − 3.31·14-s + 16-s − 3.00·17-s + 7.15·19-s + 4.42·20-s + 2.02·22-s + 14.5·25-s + 3.02·26-s − 3.31·28-s − 1.52·29-s − 5.78·31-s + 32-s − 3.00·34-s − 14.6·35-s + 1.84·37-s + 7.15·38-s + 4.42·40-s + 1.06·41-s + 1.88·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.97·5-s − 1.25·7-s + 0.353·8-s + 1.39·10-s + 0.609·11-s + 0.839·13-s − 0.886·14-s + 0.250·16-s − 0.729·17-s + 1.64·19-s + 0.989·20-s + 0.430·22-s + 2.91·25-s + 0.593·26-s − 0.626·28-s − 0.283·29-s − 1.03·31-s + 0.176·32-s − 0.515·34-s − 2.48·35-s + 0.303·37-s + 1.16·38-s + 0.699·40-s + 0.166·41-s + 0.288·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.134908869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.134908869\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 4.42T + 5T^{2} \) |
| 7 | \( 1 + 3.31T + 7T^{2} \) |
| 11 | \( 1 - 2.02T + 11T^{2} \) |
| 13 | \( 1 - 3.02T + 13T^{2} \) |
| 17 | \( 1 + 3.00T + 17T^{2} \) |
| 19 | \( 1 - 7.15T + 19T^{2} \) |
| 29 | \( 1 + 1.52T + 29T^{2} \) |
| 31 | \( 1 + 5.78T + 31T^{2} \) |
| 37 | \( 1 - 1.84T + 37T^{2} \) |
| 41 | \( 1 - 1.06T + 41T^{2} \) |
| 43 | \( 1 - 1.88T + 43T^{2} \) |
| 47 | \( 1 - 0.748T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 + 1.79T + 59T^{2} \) |
| 61 | \( 1 + 3.90T + 61T^{2} \) |
| 67 | \( 1 - 3.70T + 67T^{2} \) |
| 71 | \( 1 - 4.00T + 71T^{2} \) |
| 73 | \( 1 + 5.40T + 73T^{2} \) |
| 79 | \( 1 + 7.86T + 79T^{2} \) |
| 83 | \( 1 + 4.03T + 83T^{2} \) |
| 89 | \( 1 + 5.59T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−7.23574970635293496012216598927, −6.83209737930168378386057931282, −6.11447227730855673547881008797, −5.77260207089560055093208084378, −5.20105482892269740483182439806, −4.14940670571336667556188894098, −3.32682310672670678647824548222, −2.72072220750600791277381005709, −1.85431252117678259237756174802, −1.02879679697240430922523333284,
1.02879679697240430922523333284, 1.85431252117678259237756174802, 2.72072220750600791277381005709, 3.32682310672670678647824548222, 4.14940670571336667556188894098, 5.20105482892269740483182439806, 5.77260207089560055093208084378, 6.11447227730855673547881008797, 6.83209737930168378386057931282, 7.23574970635293496012216598927
