| L(s) = 1 | − 0.722·2-s + 0.308·3-s − 1.47·4-s − 3.27·5-s − 0.223·6-s − 2.68·7-s + 2.51·8-s − 2.90·9-s + 2.36·10-s − 3.07·11-s − 0.456·12-s + 6.97·13-s + 1.93·14-s − 1.01·15-s + 1.13·16-s − 5.53·17-s + 2.09·18-s − 5.10·19-s + 4.84·20-s − 0.828·21-s + 2.22·22-s − 7.75·23-s + 0.776·24-s + 5.74·25-s − 5.04·26-s − 1.82·27-s + 3.96·28-s + ⋯ |
| L(s) = 1 | − 0.511·2-s + 0.178·3-s − 0.738·4-s − 1.46·5-s − 0.0911·6-s − 1.01·7-s + 0.888·8-s − 0.968·9-s + 0.749·10-s − 0.926·11-s − 0.131·12-s + 1.93·13-s + 0.517·14-s − 0.261·15-s + 0.284·16-s − 1.34·17-s + 0.494·18-s − 1.17·19-s + 1.08·20-s − 0.180·21-s + 0.473·22-s − 1.61·23-s + 0.158·24-s + 1.14·25-s − 0.988·26-s − 0.351·27-s + 0.748·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3336619103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3336619103\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 983 | \( 1 - T \) |
| good | 2 | \( 1 + 0.722T + 2T^{2} \) |
| 3 | \( 1 - 0.308T + 3T^{2} \) |
| 5 | \( 1 + 3.27T + 5T^{2} \) |
| 7 | \( 1 + 2.68T + 7T^{2} \) |
| 11 | \( 1 + 3.07T + 11T^{2} \) |
| 13 | \( 1 - 6.97T + 13T^{2} \) |
| 17 | \( 1 + 5.53T + 17T^{2} \) |
| 19 | \( 1 + 5.10T + 19T^{2} \) |
| 23 | \( 1 + 7.75T + 23T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 31 | \( 1 + 0.232T + 31T^{2} \) |
| 37 | \( 1 - 9.95T + 37T^{2} \) |
| 41 | \( 1 - 1.68T + 41T^{2} \) |
| 43 | \( 1 - 1.96T + 43T^{2} \) |
| 47 | \( 1 - 2.41T + 47T^{2} \) |
| 53 | \( 1 - 4.21T + 53T^{2} \) |
| 59 | \( 1 - 8.29T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 + 3.60T + 67T^{2} \) |
| 71 | \( 1 + 9.33T + 71T^{2} \) |
| 73 | \( 1 - 2.36T + 73T^{2} \) |
| 79 | \( 1 + 4.89T + 79T^{2} \) |
| 83 | \( 1 - 1.83T + 83T^{2} \) |
| 89 | \( 1 + 7.99T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−9.981650504412831443542043394458, −8.787647688228485972774081099813, −8.419126764316037878394639011438, −7.959512438620756260301129271253, −6.64217162511663717767843359298, −5.83547942169730984051803962976, −4.30929910339903201162988239955, −3.88685623486095593270522163237, −2.69271082599046078069151340886, −0.46100811994926618251180689009,
0.46100811994926618251180689009, 2.69271082599046078069151340886, 3.88685623486095593270522163237, 4.30929910339903201162988239955, 5.83547942169730984051803962976, 6.64217162511663717767843359298, 7.959512438620756260301129271253, 8.419126764316037878394639011438, 8.787647688228485972774081099813, 9.981650504412831443542043394458
