| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 21.1·5-s − 6·6-s + 34.5·7-s − 8·8-s + 9·9-s − 42.3·10-s + 34.6·11-s + 12·12-s + 41.8·13-s − 69.0·14-s + 63.5·15-s + 16·16-s − 36.0·17-s − 18·18-s + 84.7·20-s + 103.·21-s − 69.3·22-s − 129.·23-s − 24·24-s + 323.·25-s − 83.6·26-s + 27·27-s + 138.·28-s + 193.·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.89·5-s − 0.408·6-s + 1.86·7-s − 0.353·8-s + 0.333·9-s − 1.33·10-s + 0.950·11-s + 0.288·12-s + 0.891·13-s − 1.31·14-s + 1.09·15-s + 0.250·16-s − 0.514·17-s − 0.235·18-s + 0.947·20-s + 1.07·21-s − 0.672·22-s − 1.16·23-s − 0.204·24-s + 2.58·25-s − 0.630·26-s + 0.192·27-s + 0.932·28-s + 1.23·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.525249400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.525249400\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 21.1T + 125T^{2} \) |
| 7 | \( 1 - 34.5T + 343T^{2} \) |
| 11 | \( 1 - 34.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 41.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 36.0T + 4.91e3T^{2} \) |
| 23 | \( 1 + 129.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 193.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 287.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 382.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 51.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 218.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 150.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 112.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 492.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 258.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 219.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 24.1T + 3.57e5T^{2} \) |
| 73 | \( 1 + 174.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 923.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 863.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 344.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 270.T + 9.12e5T^{2} \) |
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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
−8.696058955091734063745675601863, −8.350763520175351718655137981418, −7.20877119706294795713064800538, −6.45448044645184420884365757685, −5.64566554032202573048940381809, −4.85520387191955206673201463628, −3.71158506387702998397105086804, −2.27193726648337547331849534978, −1.74600408241521055295044186370, −1.21170654292811099517540095900,
1.21170654292811099517540095900, 1.74600408241521055295044186370, 2.27193726648337547331849534978, 3.71158506387702998397105086804, 4.85520387191955206673201463628, 5.64566554032202573048940381809, 6.45448044645184420884365757685, 7.20877119706294795713064800538, 8.350763520175351718655137981418, 8.696058955091734063745675601863
