| L(s) = 1 | + 3-s − 5-s + 2.37·7-s + 9-s − 0.372·11-s + 6.74·13-s − 15-s + 17-s + 2.37·19-s + 2.37·21-s + 4·23-s + 25-s + 27-s − 8.37·29-s + 10.7·31-s − 0.372·33-s − 2.37·35-s − 4.37·37-s + 6.74·39-s − 8.37·41-s − 2.74·43-s − 45-s − 0.372·47-s − 1.37·49-s + 51-s − 0.372·53-s + 0.372·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.896·7-s + 0.333·9-s − 0.112·11-s + 1.87·13-s − 0.258·15-s + 0.242·17-s + 0.544·19-s + 0.517·21-s + 0.834·23-s + 0.200·25-s + 0.192·27-s − 1.55·29-s + 1.92·31-s − 0.0648·33-s − 0.400·35-s − 0.718·37-s + 1.07·39-s − 1.30·41-s − 0.418·43-s − 0.149·45-s − 0.0543·47-s − 0.196·49-s + 0.140·51-s − 0.0511·53-s + 0.0501·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.855727240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.855727240\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 2.37T + 7T^{2} \) |
| 11 | \( 1 + 0.372T + 11T^{2} \) |
| 13 | \( 1 - 6.74T + 13T^{2} \) |
| 19 | \( 1 - 2.37T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 8.37T + 29T^{2} \) |
| 31 | \( 1 - 10.7T + 31T^{2} \) |
| 37 | \( 1 + 4.37T + 37T^{2} \) |
| 41 | \( 1 + 8.37T + 41T^{2} \) |
| 43 | \( 1 + 2.74T + 43T^{2} \) |
| 47 | \( 1 + 0.372T + 47T^{2} \) |
| 53 | \( 1 + 0.372T + 53T^{2} \) |
| 59 | \( 1 - 3.25T + 59T^{2} \) |
| 61 | \( 1 + 1.25T + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 - 1.25T + 71T^{2} \) |
| 73 | \( 1 - 8.37T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 6.74T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−8.435351726876670139307015539618, −7.88339759916068175098963221216, −7.10353947298743953839431948333, −6.29432358965993950226331175715, −5.35438855810345216657099297732, −4.62071643035494242295630794022, −3.67943718174055574225296457129, −3.16185324677643153204082618303, −1.85326577289203764772748250484, −1.02532679188539279013011532355,
1.02532679188539279013011532355, 1.85326577289203764772748250484, 3.16185324677643153204082618303, 3.67943718174055574225296457129, 4.62071643035494242295630794022, 5.35438855810345216657099297732, 6.29432358965993950226331175715, 7.10353947298743953839431948333, 7.88339759916068175098963221216, 8.435351726876670139307015539618
