| L(s) = 1 | + 2-s − 3-s + 4-s + 3.34·5-s − 6-s + 0.833·7-s + 8-s + 9-s + 3.34·10-s + 11-s − 12-s + 2.47·13-s + 0.833·14-s − 3.34·15-s + 16-s + 5.78·17-s + 18-s + 2.48·19-s + 3.34·20-s − 0.833·21-s + 22-s − 8.08·23-s − 24-s + 6.21·25-s + 2.47·26-s − 27-s + 0.833·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.49·5-s − 0.408·6-s + 0.314·7-s + 0.353·8-s + 0.333·9-s + 1.05·10-s + 0.301·11-s − 0.288·12-s + 0.687·13-s + 0.222·14-s − 0.864·15-s + 0.250·16-s + 1.40·17-s + 0.235·18-s + 0.570·19-s + 0.748·20-s − 0.181·21-s + 0.213·22-s − 1.68·23-s − 0.204·24-s + 1.24·25-s + 0.486·26-s − 0.192·27-s + 0.157·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.980242762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.980242762\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 5 | \( 1 - 3.34T + 5T^{2} \) |
| 7 | \( 1 - 0.833T + 7T^{2} \) |
| 13 | \( 1 - 2.47T + 13T^{2} \) |
| 17 | \( 1 - 5.78T + 17T^{2} \) |
| 19 | \( 1 - 2.48T + 19T^{2} \) |
| 23 | \( 1 + 8.08T + 23T^{2} \) |
| 29 | \( 1 - 7.61T + 29T^{2} \) |
| 31 | \( 1 + 4.82T + 31T^{2} \) |
| 37 | \( 1 - 1.55T + 37T^{2} \) |
| 41 | \( 1 - 5.40T + 41T^{2} \) |
| 43 | \( 1 - 2.31T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 + 6.62T + 53T^{2} \) |
| 59 | \( 1 - 4.56T + 59T^{2} \) |
| 67 | \( 1 - 2.65T + 67T^{2} \) |
| 71 | \( 1 - 5.22T + 71T^{2} \) |
| 73 | \( 1 + 3.14T + 73T^{2} \) |
| 79 | \( 1 + 6.92T + 79T^{2} \) |
| 83 | \( 1 + 0.942T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−8.314390329311966482609205297952, −7.62833063075925924618569072562, −6.54254868030054839078250799401, −6.11638692205100627593177678557, −5.53539836943969952649275599932, −4.91977418794607829188825584376, −3.92480523196346363651629332528, −2.99485979106395518628171341060, −1.87530449815245068111317318944, −1.19696192855130500725328902994,
1.19696192855130500725328902994, 1.87530449815245068111317318944, 2.99485979106395518628171341060, 3.92480523196346363651629332528, 4.91977418794607829188825584376, 5.53539836943969952649275599932, 6.11638692205100627593177678557, 6.54254868030054839078250799401, 7.62833063075925924618569072562, 8.314390329311966482609205297952
