| L(s) = 1 | + 2-s + 3-s + 4-s − 1.01·5-s + 6-s − 2.46·7-s + 8-s + 9-s − 1.01·10-s − 4.82·11-s + 12-s + 2.72·13-s − 2.46·14-s − 1.01·15-s + 16-s − 17-s + 18-s + 2.53·19-s − 1.01·20-s − 2.46·21-s − 4.82·22-s − 2.05·23-s + 24-s − 3.97·25-s + 2.72·26-s + 27-s − 2.46·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.452·5-s + 0.408·6-s − 0.932·7-s + 0.353·8-s + 0.333·9-s − 0.319·10-s − 1.45·11-s + 0.288·12-s + 0.755·13-s − 0.659·14-s − 0.261·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s + 0.581·19-s − 0.226·20-s − 0.538·21-s − 1.02·22-s − 0.429·23-s + 0.204·24-s − 0.795·25-s + 0.534·26-s + 0.192·27-s − 0.466·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.968947217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.968947217\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 + 1.01T + 5T^{2} \) |
| 7 | \( 1 + 2.46T + 7T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 - 2.72T + 13T^{2} \) |
| 19 | \( 1 - 2.53T + 19T^{2} \) |
| 23 | \( 1 + 2.05T + 23T^{2} \) |
| 29 | \( 1 - 8.41T + 29T^{2} \) |
| 31 | \( 1 - 6.58T + 31T^{2} \) |
| 37 | \( 1 - 5.59T + 37T^{2} \) |
| 41 | \( 1 - 6.08T + 41T^{2} \) |
| 43 | \( 1 + 8.91T + 43T^{2} \) |
| 47 | \( 1 - 6.21T + 47T^{2} \) |
| 53 | \( 1 - 3.60T + 53T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 4.45T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 + 1.79T + 73T^{2} \) |
| 79 | \( 1 + 0.511T + 79T^{2} \) |
| 83 | \( 1 - 5.01T + 83T^{2} \) |
| 89 | \( 1 - 6.59T + 89T^{2} \) |
| 97 | \( 1 - 4.95T + 97T^{2} \) |
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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
−8.083489844869246573795737759212, −7.39354765169519565350140405122, −6.56596430695934727777159516911, −5.99433798411698659968045069638, −5.12315624452869057431452247859, −4.32830192126382384337053121080, −3.58126972483922860339009732218, −2.89946183141676092885615343152, −2.28235022087887955860722183651, −0.76966994480678243795231189134,
0.76966994480678243795231189134, 2.28235022087887955860722183651, 2.89946183141676092885615343152, 3.58126972483922860339009732218, 4.32830192126382384337053121080, 5.12315624452869057431452247859, 5.99433798411698659968045069638, 6.56596430695934727777159516911, 7.39354765169519565350140405122, 8.083489844869246573795737759212
