| L(s) = 1 | + 1.43·2-s + 3-s + 0.0731·4-s + 0.926·5-s + 1.43·6-s + 7-s − 2.77·8-s + 9-s + 1.33·10-s + 4.21·11-s + 0.0731·12-s + 13-s + 1.43·14-s + 0.926·15-s − 4.14·16-s − 2.87·17-s + 1.43·18-s − 1.28·19-s + 0.0678·20-s + 21-s + 6.06·22-s − 8.02·23-s − 2.77·24-s − 4.14·25-s + 1.43·26-s + 27-s + 0.0731·28-s + ⋯ |
| L(s) = 1 | + 1.01·2-s + 0.577·3-s + 0.0365·4-s + 0.414·5-s + 0.587·6-s + 0.377·7-s − 0.980·8-s + 0.333·9-s + 0.422·10-s + 1.27·11-s + 0.0211·12-s + 0.277·13-s + 0.384·14-s + 0.239·15-s − 1.03·16-s − 0.698·17-s + 0.339·18-s − 0.295·19-s + 0.0151·20-s + 0.218·21-s + 1.29·22-s − 1.67·23-s − 0.566·24-s − 0.828·25-s + 0.282·26-s + 0.192·27-s + 0.0138·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.378511152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.378511152\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 1.43T + 2T^{2} \) |
| 5 | \( 1 - 0.926T + 5T^{2} \) |
| 11 | \( 1 - 4.21T + 11T^{2} \) |
| 17 | \( 1 + 2.87T + 17T^{2} \) |
| 19 | \( 1 + 1.28T + 19T^{2} \) |
| 23 | \( 1 + 8.02T + 23T^{2} \) |
| 29 | \( 1 - 3.28T + 29T^{2} \) |
| 31 | \( 1 + 7.04T + 31T^{2} \) |
| 37 | \( 1 - 8.57T + 37T^{2} \) |
| 41 | \( 1 + 12.0T + 41T^{2} \) |
| 43 | \( 1 - 7.14T + 43T^{2} \) |
| 47 | \( 1 - 1.95T + 47T^{2} \) |
| 53 | \( 1 + 5.14T + 53T^{2} \) |
| 59 | \( 1 + 7.33T + 59T^{2} \) |
| 61 | \( 1 - 7.75T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 8.32T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 + 3.80T + 83T^{2} \) |
| 89 | \( 1 + 5.64T + 89T^{2} \) |
| 97 | \( 1 - 6.90T + 97T^{2} \) |
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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
−12.10981021005820956210283167608, −11.26086739905558096152940656931, −9.835827071223691262533351558575, −9.065664092473384805290368396835, −8.140897107322258190302295179480, −6.64240652262746016190324874885, −5.79326950310041801455857676961, −4.42163015107767800794202903827, −3.68341177093957050134082259414, −2.06303734463239796355681587821,
2.06303734463239796355681587821, 3.68341177093957050134082259414, 4.42163015107767800794202903827, 5.79326950310041801455857676961, 6.64240652262746016190324874885, 8.140897107322258190302295179480, 9.065664092473384805290368396835, 9.835827071223691262533351558575, 11.26086739905558096152940656931, 12.10981021005820956210283167608
