| L(s) = 1 | − 2-s + 3-s + 4-s − 1.19·5-s − 6-s − 0.609·7-s − 8-s + 9-s + 1.19·10-s − 11-s + 12-s − 4.29·13-s + 0.609·14-s − 1.19·15-s + 16-s + 4.81·17-s − 18-s − 0.143·19-s − 1.19·20-s − 0.609·21-s + 22-s + 5.45·23-s − 24-s − 3.58·25-s + 4.29·26-s + 27-s − 0.609·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.532·5-s − 0.408·6-s − 0.230·7-s − 0.353·8-s + 0.333·9-s + 0.376·10-s − 0.301·11-s + 0.288·12-s − 1.19·13-s + 0.162·14-s − 0.307·15-s + 0.250·16-s + 1.16·17-s − 0.235·18-s − 0.0329·19-s − 0.266·20-s − 0.132·21-s + 0.213·22-s + 1.13·23-s − 0.204·24-s − 0.716·25-s + 0.842·26-s + 0.192·27-s − 0.115·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 5 | \( 1 + 1.19T + 5T^{2} \) |
| 7 | \( 1 + 0.609T + 7T^{2} \) |
| 13 | \( 1 + 4.29T + 13T^{2} \) |
| 17 | \( 1 - 4.81T + 17T^{2} \) |
| 19 | \( 1 + 0.143T + 19T^{2} \) |
| 23 | \( 1 - 5.45T + 23T^{2} \) |
| 29 | \( 1 + 7.48T + 29T^{2} \) |
| 31 | \( 1 - 7.52T + 31T^{2} \) |
| 37 | \( 1 - 3.16T + 37T^{2} \) |
| 41 | \( 1 + 6.53T + 41T^{2} \) |
| 43 | \( 1 - 0.559T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 67 | \( 1 + 3.04T + 67T^{2} \) |
| 71 | \( 1 + 9.92T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 + 0.542T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 + 8.59T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−8.053349428803421458986430399162, −7.45501156764850793687649113459, −7.03920963807978222229039357776, −5.92902107815683103069277943554, −5.09265764639586087322230937848, −4.12574804136113949822176384466, −3.16611622088774209484239955441, −2.54247979160432359501931285506, −1.34375846112533918275294567095, 0,
1.34375846112533918275294567095, 2.54247979160432359501931285506, 3.16611622088774209484239955441, 4.12574804136113949822176384466, 5.09265764639586087322230937848, 5.92902107815683103069277943554, 7.03920963807978222229039357776, 7.45501156764850793687649113459, 8.053349428803421458986430399162
