| L(s) = 1 | + 2-s + 2·3-s + 4-s + 5-s + 2·6-s − 4.37·7-s + 8-s + 9-s + 10-s + 2.37·11-s + 2·12-s + 6.74·13-s − 4.37·14-s + 2·15-s + 16-s + 0.372·17-s + 18-s − 2·19-s + 20-s − 8.74·21-s + 2.37·22-s − 4.74·23-s + 2·24-s + 25-s + 6.74·26-s − 4·27-s − 4.37·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 0.5·4-s + 0.447·5-s + 0.816·6-s − 1.65·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.715·11-s + 0.577·12-s + 1.87·13-s − 1.16·14-s + 0.516·15-s + 0.250·16-s + 0.0902·17-s + 0.235·18-s − 0.458·19-s + 0.223·20-s − 1.90·21-s + 0.505·22-s − 0.989·23-s + 0.408·24-s + 0.200·25-s + 1.32·26-s − 0.769·27-s − 0.826·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.710486559\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.710486559\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 - 2T + 3T^{2} \) |
| 7 | \( 1 + 4.37T + 7T^{2} \) |
| 11 | \( 1 - 2.37T + 11T^{2} \) |
| 13 | \( 1 - 6.74T + 13T^{2} \) |
| 17 | \( 1 - 0.372T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 4.74T + 23T^{2} \) |
| 29 | \( 1 + 9.11T + 29T^{2} \) |
| 31 | \( 1 + 8.37T + 31T^{2} \) |
| 41 | \( 1 + 0.372T + 41T^{2} \) |
| 43 | \( 1 - 1.62T + 43T^{2} \) |
| 47 | \( 1 - 2.74T + 47T^{2} \) |
| 53 | \( 1 + 4.37T + 53T^{2} \) |
| 59 | \( 1 - 1.25T + 59T^{2} \) |
| 61 | \( 1 - 0.372T + 61T^{2} \) |
| 67 | \( 1 + 6.74T + 67T^{2} \) |
| 71 | \( 1 - 4.74T + 71T^{2} \) |
| 73 | \( 1 + 2.74T + 73T^{2} \) |
| 79 | \( 1 - 6.74T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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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
−11.46613437349725801106723157171, −10.43111387338297534143893299370, −9.312317635453180027617770363635, −8.895229980931036036550921747096, −7.57899665809056725753882985405, −6.34714050316419728421997080073, −5.87169576061093445842092703018, −3.74786059982972828851228147626, −3.49455304925544319783485761219, −2.02259618606569259686588845236,
2.02259618606569259686588845236, 3.49455304925544319783485761219, 3.74786059982972828851228147626, 5.87169576061093445842092703018, 6.34714050316419728421997080073, 7.57899665809056725753882985405, 8.895229980931036036550921747096, 9.312317635453180027617770363635, 10.43111387338297534143893299370, 11.46613437349725801106723157171
