| L(s) = 1 | − 2-s − 2.35·3-s + 4-s − 3.94·5-s + 2.35·6-s − 0.732·7-s − 8-s + 2.52·9-s + 3.94·10-s − 5.60·11-s − 2.35·12-s − 6.57·13-s + 0.732·14-s + 9.27·15-s + 16-s + 7.88·17-s − 2.52·18-s − 3.58·19-s − 3.94·20-s + 1.72·21-s + 5.60·22-s + 0.597·23-s + 2.35·24-s + 10.5·25-s + 6.57·26-s + 1.10·27-s − 0.732·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.35·3-s + 0.5·4-s − 1.76·5-s + 0.959·6-s − 0.276·7-s − 0.353·8-s + 0.843·9-s + 1.24·10-s − 1.68·11-s − 0.678·12-s − 1.82·13-s + 0.195·14-s + 2.39·15-s + 0.250·16-s + 1.91·17-s − 0.596·18-s − 0.823·19-s − 0.882·20-s + 0.375·21-s + 1.19·22-s + 0.124·23-s + 0.479·24-s + 2.11·25-s + 1.28·26-s + 0.213·27-s − 0.138·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 + 2.35T + 3T^{2} \) |
| 5 | \( 1 + 3.94T + 5T^{2} \) |
| 7 | \( 1 + 0.732T + 7T^{2} \) |
| 11 | \( 1 + 5.60T + 11T^{2} \) |
| 13 | \( 1 + 6.57T + 13T^{2} \) |
| 17 | \( 1 - 7.88T + 17T^{2} \) |
| 19 | \( 1 + 3.58T + 19T^{2} \) |
| 23 | \( 1 - 0.597T + 23T^{2} \) |
| 29 | \( 1 + 5.16T + 29T^{2} \) |
| 31 | \( 1 - 2.38T + 31T^{2} \) |
| 37 | \( 1 + 8.53T + 37T^{2} \) |
| 41 | \( 1 - 5.16T + 41T^{2} \) |
| 43 | \( 1 - 4.23T + 43T^{2} \) |
| 47 | \( 1 + 2.11T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 - 14.6T + 59T^{2} \) |
| 61 | \( 1 - 3.15T + 61T^{2} \) |
| 67 | \( 1 - 15.0T + 67T^{2} \) |
| 71 | \( 1 + 0.278T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 + 3.94T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 - 1.59T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−7.984313920408714653752704841554, −7.19143964367191565423102392781, −7.12056145218347059071564013351, −5.66581185340510552772954343730, −5.26294545094758117932004640434, −4.43723453467826524199545713296, −3.36067351300620503720321033066, −2.47200801503448274155302249943, −0.66432416121922689682857374735, 0,
0.66432416121922689682857374735, 2.47200801503448274155302249943, 3.36067351300620503720321033066, 4.43723453467826524199545713296, 5.26294545094758117932004640434, 5.66581185340510552772954343730, 7.12056145218347059071564013351, 7.19143964367191565423102392781, 7.984313920408714653752704841554
