| L(s) = 1 | − 2-s − 3-s + 4-s + 3.53·5-s + 6-s − 0.614·7-s − 8-s + 9-s − 3.53·10-s + 0.607·11-s − 12-s − 5.13·13-s + 0.614·14-s − 3.53·15-s + 16-s − 4.89·17-s − 18-s + 4.89·19-s + 3.53·20-s + 0.614·21-s − 0.607·22-s − 23-s + 24-s + 7.46·25-s + 5.13·26-s − 27-s − 0.614·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.57·5-s + 0.408·6-s − 0.232·7-s − 0.353·8-s + 0.333·9-s − 1.11·10-s + 0.183·11-s − 0.288·12-s − 1.42·13-s + 0.164·14-s − 0.911·15-s + 0.250·16-s − 1.18·17-s − 0.235·18-s + 1.12·19-s + 0.789·20-s + 0.134·21-s − 0.129·22-s − 0.208·23-s + 0.204·24-s + 1.49·25-s + 1.00·26-s − 0.192·27-s − 0.116·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 3.53T + 5T^{2} \) |
| 7 | \( 1 + 0.614T + 7T^{2} \) |
| 11 | \( 1 - 0.607T + 11T^{2} \) |
| 13 | \( 1 + 5.13T + 13T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 - 4.89T + 19T^{2} \) |
| 31 | \( 1 + 0.273T + 31T^{2} \) |
| 37 | \( 1 + 5.09T + 37T^{2} \) |
| 41 | \( 1 - 5.34T + 41T^{2} \) |
| 43 | \( 1 + 1.78T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 1.81T + 53T^{2} \) |
| 59 | \( 1 + 2.87T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 + 7.92T + 71T^{2} \) |
| 73 | \( 1 + 6.01T + 73T^{2} \) |
| 79 | \( 1 - 0.712T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 5.81T + 97T^{2} \) |
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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
−8.126080078845210706096777684794, −7.11767587075958530621208366418, −6.73510098975930028855642597122, −5.90823081709160855108496348752, −5.29863057174063296530974905829, −4.53573558772942794330289448696, −3.06714987111835461054128383817, −2.21055314502806863532332700126, −1.44490116772044788147592720331, 0,
1.44490116772044788147592720331, 2.21055314502806863532332700126, 3.06714987111835461054128383817, 4.53573558772942794330289448696, 5.29863057174063296530974905829, 5.90823081709160855108496348752, 6.73510098975930028855642597122, 7.11767587075958530621208366418, 8.126080078845210706096777684794
