| L(s) = 1 | − 2-s − 3-s + 4-s + 1.38·5-s + 6-s + 2.61·7-s − 8-s + 9-s − 1.38·10-s + 0.381·11-s − 12-s − 4.47·13-s − 2.61·14-s − 1.38·15-s + 16-s + 0.763·17-s − 18-s + 1.38·20-s − 2.61·21-s − 0.381·22-s + 2.47·23-s + 24-s − 3.09·25-s + 4.47·26-s − 27-s + 2.61·28-s + 7.61·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.618·5-s + 0.408·6-s + 0.989·7-s − 0.353·8-s + 0.333·9-s − 0.437·10-s + 0.115·11-s − 0.288·12-s − 1.24·13-s − 0.699·14-s − 0.356·15-s + 0.250·16-s + 0.185·17-s − 0.235·18-s + 0.309·20-s − 0.571·21-s − 0.0814·22-s + 0.515·23-s + 0.204·24-s − 0.618·25-s + 0.877·26-s − 0.192·27-s + 0.494·28-s + 1.41·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.294227212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.294227212\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 1.38T + 5T^{2} \) |
| 7 | \( 1 - 2.61T + 7T^{2} \) |
| 11 | \( 1 - 0.381T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 0.763T + 17T^{2} \) |
| 23 | \( 1 - 2.47T + 23T^{2} \) |
| 29 | \( 1 - 7.61T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 1.70T + 41T^{2} \) |
| 43 | \( 1 - 3.52T + 43T^{2} \) |
| 47 | \( 1 + 10T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 + 5.14T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 - 8.94T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 - 8.32T + 73T^{2} \) |
| 79 | \( 1 - 6.38T + 79T^{2} \) |
| 83 | \( 1 - 7.09T + 83T^{2} \) |
| 89 | \( 1 - 18.6T + 89T^{2} \) |
| 97 | \( 1 + 4.47T + 97T^{2} \) |
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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
−9.253537723321026715805228437915, −8.085416025936169450009873616543, −7.78894599188415890829759921514, −6.65429004480521561121852762009, −6.12769261016404721165738184364, −4.99352175496905333657675697732, −4.58263452746903447352586723117, −2.91086582695484780300211569331, −1.95864795423402318135134479697, −0.880272925384478930441283893335,
0.880272925384478930441283893335, 1.95864795423402318135134479697, 2.91086582695484780300211569331, 4.58263452746903447352586723117, 4.99352175496905333657675697732, 6.12769261016404721165738184364, 6.65429004480521561121852762009, 7.78894599188415890829759921514, 8.085416025936169450009873616543, 9.253537723321026715805228437915
