L(s) = 1 | − 2-s + 2.89·3-s + 4-s − 5-s − 2.89·6-s − 4.89·7-s − 8-s + 5.40·9-s + 10-s − 0.723·11-s + 2.89·12-s + 5.93·13-s + 4.89·14-s − 2.89·15-s + 16-s − 6.99·17-s − 5.40·18-s + 5.63·19-s − 20-s − 14.1·21-s + 0.723·22-s − 1.75·23-s − 2.89·24-s + 25-s − 5.93·26-s + 6.96·27-s − 4.89·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.67·3-s + 0.5·4-s − 0.447·5-s − 1.18·6-s − 1.85·7-s − 0.353·8-s + 1.80·9-s + 0.316·10-s − 0.218·11-s + 0.836·12-s + 1.64·13-s + 1.30·14-s − 0.748·15-s + 0.250·16-s − 1.69·17-s − 1.27·18-s + 1.29·19-s − 0.223·20-s − 3.09·21-s + 0.154·22-s − 0.366·23-s − 0.591·24-s + 0.200·25-s − 1.16·26-s + 1.33·27-s − 0.925·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.870853780\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.870853780\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 - 2.89T + 3T^{2} \) |
| 7 | \( 1 + 4.89T + 7T^{2} \) |
| 11 | \( 1 + 0.723T + 11T^{2} \) |
| 13 | \( 1 - 5.93T + 13T^{2} \) |
| 17 | \( 1 + 6.99T + 17T^{2} \) |
| 19 | \( 1 - 5.63T + 19T^{2} \) |
| 23 | \( 1 + 1.75T + 23T^{2} \) |
| 29 | \( 1 + 6.38T + 29T^{2} \) |
| 31 | \( 1 - 0.258T + 31T^{2} \) |
| 37 | \( 1 - 8.91T + 37T^{2} \) |
| 41 | \( 1 - 0.158T + 41T^{2} \) |
| 43 | \( 1 - 7.31T + 43T^{2} \) |
| 47 | \( 1 - 9.12T + 47T^{2} \) |
| 53 | \( 1 - 2.96T + 53T^{2} \) |
| 59 | \( 1 + 4.56T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 - 9.21T + 67T^{2} \) |
| 71 | \( 1 + 7.78T + 71T^{2} \) |
| 73 | \( 1 - 0.261T + 73T^{2} \) |
| 79 | \( 1 + 4.33T + 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 - 6.98T + 89T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
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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
−8.633660298501416862090417971467, −7.81875697359103191853575799394, −7.23838745088457789304972124392, −6.51341729708655488370507716297, −5.80187820421707651020480282897, −4.06826245739096290638583435694, −3.67315786647544706756176562788, −2.89923789934064387490746977602, −2.20810231754141294600767759798, −0.78694003023072985612545256304,
0.78694003023072985612545256304, 2.20810231754141294600767759798, 2.89923789934064387490746977602, 3.67315786647544706756176562788, 4.06826245739096290638583435694, 5.80187820421707651020480282897, 6.51341729708655488370507716297, 7.23838745088457789304972124392, 7.81875697359103191853575799394, 8.633660298501416862090417971467