| L(s) = 1 | + 2.35·2-s − 0.395·3-s − 2.45·4-s − 11.6·5-s − 0.931·6-s + 7.76·7-s − 24.6·8-s − 26.8·9-s − 27.4·10-s + 22.2·11-s + 0.970·12-s − 72.1·13-s + 18.2·14-s + 4.60·15-s − 38.3·16-s + 43.4·17-s − 63.2·18-s + 43.1·19-s + 28.5·20-s − 3.07·21-s + 52.3·22-s − 19.6·23-s + 9.73·24-s + 10.5·25-s − 170.·26-s + 21.2·27-s − 19.0·28-s + ⋯ |
| L(s) = 1 | + 0.832·2-s − 0.0761·3-s − 0.306·4-s − 1.04·5-s − 0.0633·6-s + 0.419·7-s − 1.08·8-s − 0.994·9-s − 0.867·10-s + 0.609·11-s + 0.0233·12-s − 1.53·13-s + 0.349·14-s + 0.0792·15-s − 0.599·16-s + 0.620·17-s − 0.827·18-s + 0.520·19-s + 0.319·20-s − 0.0319·21-s + 0.507·22-s − 0.178·23-s + 0.0828·24-s + 0.0846·25-s − 1.28·26-s + 0.151·27-s − 0.128·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.324267452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.324267452\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 \) |
| good | 2 | \( 1 - 2.35T + 8T^{2} \) |
| 3 | \( 1 + 0.395T + 27T^{2} \) |
| 5 | \( 1 + 11.6T + 125T^{2} \) |
| 7 | \( 1 - 7.76T + 343T^{2} \) |
| 11 | \( 1 - 22.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 72.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 43.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 43.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 19.6T + 1.21e4T^{2} \) |
| 31 | \( 1 + 202.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 169.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 227.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 500.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 431.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 206.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 163.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 25.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 806.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.01e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 579.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 101.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.07e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 717.T + 9.12e5T^{2} \) |
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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
−9.628045714850246176443718349098, −9.019578945615186370070511787894, −7.948566920512214736515150273368, −7.37510821933361905483842734237, −6.01727452320689711563788138014, −5.25061652340347404084644580295, −4.39339887484256382695949541716, −3.58029578311543974589038097276, −2.55953103277961800841817375328, −0.54550859261316443324030122985,
0.54550859261316443324030122985, 2.55953103277961800841817375328, 3.58029578311543974589038097276, 4.39339887484256382695949541716, 5.25061652340347404084644580295, 6.01727452320689711563788138014, 7.37510821933361905483842734237, 7.948566920512214736515150273368, 9.019578945615186370070511787894, 9.628045714850246176443718349098
