| L(s) = 1 | + 2.73·2-s + 3-s + 5.45·4-s + 5-s + 2.73·6-s + 0.454·7-s + 9.44·8-s + 9-s + 2.73·10-s − 3.84·11-s + 5.45·12-s + 1.79·13-s + 1.24·14-s + 15-s + 14.8·16-s + 1.04·17-s + 2.73·18-s + 4.39·19-s + 5.45·20-s + 0.454·21-s − 10.5·22-s − 3.13·23-s + 9.44·24-s + 25-s + 4.88·26-s + 27-s + 2.48·28-s + ⋯ |
| L(s) = 1 | + 1.93·2-s + 0.577·3-s + 2.72·4-s + 0.447·5-s + 1.11·6-s + 0.171·7-s + 3.33·8-s + 0.333·9-s + 0.863·10-s − 1.16·11-s + 1.57·12-s + 0.496·13-s + 0.331·14-s + 0.258·15-s + 3.71·16-s + 0.252·17-s + 0.643·18-s + 1.00·19-s + 1.22·20-s + 0.0992·21-s − 2.24·22-s − 0.653·23-s + 1.92·24-s + 0.200·25-s + 0.958·26-s + 0.192·27-s + 0.469·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.40262219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.40262219\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 2.73T + 2T^{2} \) |
| 7 | \( 1 - 0.454T + 7T^{2} \) |
| 11 | \( 1 + 3.84T + 11T^{2} \) |
| 13 | \( 1 - 1.79T + 13T^{2} \) |
| 17 | \( 1 - 1.04T + 17T^{2} \) |
| 19 | \( 1 - 4.39T + 19T^{2} \) |
| 23 | \( 1 + 3.13T + 23T^{2} \) |
| 29 | \( 1 - 0.178T + 29T^{2} \) |
| 31 | \( 1 - 8.66T + 31T^{2} \) |
| 37 | \( 1 - 0.404T + 37T^{2} \) |
| 41 | \( 1 + 9.16T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 - 5.34T + 53T^{2} \) |
| 59 | \( 1 + 4.88T + 59T^{2} \) |
| 61 | \( 1 + 1.72T + 61T^{2} \) |
| 67 | \( 1 - 6.24T + 67T^{2} \) |
| 71 | \( 1 - 3.18T + 71T^{2} \) |
| 73 | \( 1 - 5.08T + 73T^{2} \) |
| 79 | \( 1 - 6.02T + 79T^{2} \) |
| 83 | \( 1 + 5.09T + 83T^{2} \) |
| 89 | \( 1 + 15.0T + 89T^{2} \) |
| 97 | \( 1 + 4.60T + 97T^{2} \) |
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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
−8.047499108112874084933062002569, −7.07953390568781907671121158147, −6.49143404925110256167100614150, −5.74140091883476836096246660648, −5.04895328026000612993928255264, −4.61965258043878411210522941419, −3.47185068791660886955014849447, −3.11307552291097562025629794292, −2.24127656047269623738452380795, −1.44419007465565192172289713331,
1.44419007465565192172289713331, 2.24127656047269623738452380795, 3.11307552291097562025629794292, 3.47185068791660886955014849447, 4.61965258043878411210522941419, 5.04895328026000612993928255264, 5.74140091883476836096246660648, 6.49143404925110256167100614150, 7.07953390568781907671121158147, 8.047499108112874084933062002569
