| L(s) = 1 | + 2-s − 3-s + 4-s + 2.29·5-s − 6-s − 7-s + 8-s + 9-s + 2.29·10-s − 4.18·11-s − 12-s + 2.74·13-s − 14-s − 2.29·15-s + 16-s + 1.08·17-s + 18-s + 1.38·19-s + 2.29·20-s + 21-s − 4.18·22-s − 3.41·23-s − 24-s + 0.274·25-s + 2.74·26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.02·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.726·10-s − 1.26·11-s − 0.288·12-s + 0.761·13-s − 0.267·14-s − 0.592·15-s + 0.250·16-s + 0.264·17-s + 0.235·18-s + 0.318·19-s + 0.513·20-s + 0.218·21-s − 0.892·22-s − 0.711·23-s − 0.204·24-s + 0.0548·25-s + 0.538·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 - T \) |
| good | 5 | \( 1 - 2.29T + 5T^{2} \) |
| 11 | \( 1 + 4.18T + 11T^{2} \) |
| 13 | \( 1 - 2.74T + 13T^{2} \) |
| 17 | \( 1 - 1.08T + 17T^{2} \) |
| 19 | \( 1 - 1.38T + 19T^{2} \) |
| 23 | \( 1 + 3.41T + 23T^{2} \) |
| 29 | \( 1 + 2.29T + 29T^{2} \) |
| 31 | \( 1 + 9.91T + 31T^{2} \) |
| 37 | \( 1 - 3.46T + 37T^{2} \) |
| 41 | \( 1 + 7.29T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 + 5.40T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + 8.17T + 59T^{2} \) |
| 61 | \( 1 + 7.34T + 61T^{2} \) |
| 67 | \( 1 - 9.14T + 67T^{2} \) |
| 71 | \( 1 + 6.42T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 - 3.77T + 79T^{2} \) |
| 83 | \( 1 + 2.83T + 83T^{2} \) |
| 89 | \( 1 + 6.67T + 89T^{2} \) |
| 97 | \( 1 - 0.880T + 97T^{2} \) |
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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
−7.32947440948291861531846322456, −6.51710192764182647677533656417, −5.97390092176035752276304472447, −5.39749429896309673012490760589, −4.99687253491538237734113618984, −3.85737017542484390747891132613, −3.21421553030258538227929511414, −2.20224877669619299034565320185, −1.51847496686629782932518350110, 0,
1.51847496686629782932518350110, 2.20224877669619299034565320185, 3.21421553030258538227929511414, 3.85737017542484390747891132613, 4.99687253491538237734113618984, 5.39749429896309673012490760589, 5.97390092176035752276304472447, 6.51710192764182647677533656417, 7.32947440948291861531846322456
