| L(s) = 1 | − 1.73·3-s + 2.38·5-s + 2.01·7-s + 0.00574·9-s − 1.91·11-s − 3.34·13-s − 4.13·15-s + 17-s − 3.01·19-s − 3.49·21-s − 8.48·23-s + 0.690·25-s + 5.19·27-s + 6.27·29-s + 7.37·31-s + 3.32·33-s + 4.81·35-s + 6.24·37-s + 5.79·39-s + 9.02·41-s + 0.659·43-s + 0.0137·45-s − 0.0986·47-s − 2.92·49-s − 1.73·51-s − 1.82·53-s − 4.57·55-s + ⋯ |
| L(s) = 1 | − 1.00·3-s + 1.06·5-s + 0.762·7-s + 0.00191·9-s − 0.578·11-s − 0.927·13-s − 1.06·15-s + 0.242·17-s − 0.691·19-s − 0.763·21-s − 1.76·23-s + 0.138·25-s + 0.999·27-s + 1.16·29-s + 1.32·31-s + 0.578·33-s + 0.813·35-s + 1.02·37-s + 0.928·39-s + 1.40·41-s + 0.100·43-s + 0.00204·45-s − 0.0143·47-s − 0.417·49-s − 0.242·51-s − 0.250·53-s − 0.616·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + 1.73T + 3T^{2} \) |
| 5 | \( 1 - 2.38T + 5T^{2} \) |
| 7 | \( 1 - 2.01T + 7T^{2} \) |
| 11 | \( 1 + 1.91T + 11T^{2} \) |
| 13 | \( 1 + 3.34T + 13T^{2} \) |
| 19 | \( 1 + 3.01T + 19T^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 29 | \( 1 - 6.27T + 29T^{2} \) |
| 31 | \( 1 - 7.37T + 31T^{2} \) |
| 37 | \( 1 - 6.24T + 37T^{2} \) |
| 41 | \( 1 - 9.02T + 41T^{2} \) |
| 43 | \( 1 - 0.659T + 43T^{2} \) |
| 47 | \( 1 + 0.0986T + 47T^{2} \) |
| 53 | \( 1 + 1.82T + 53T^{2} \) |
| 61 | \( 1 + 9.20T + 61T^{2} \) |
| 67 | \( 1 - 2.99T + 67T^{2} \) |
| 71 | \( 1 + 3.65T + 71T^{2} \) |
| 73 | \( 1 + 0.700T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 - 2.46T + 89T^{2} \) |
| 97 | \( 1 - 18.8T + 97T^{2} \) |
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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
−7.56756149455693143394541194667, −6.42344517727849548057262253661, −6.09793983469196203141262840170, −5.50170953376952136648421110233, −4.75087494207642207202016603156, −4.31647844119580689503729587037, −2.76570808187984056604119583920, −2.25730737832222471083525645577, −1.21239960906130905259829110070, 0,
1.21239960906130905259829110070, 2.25730737832222471083525645577, 2.76570808187984056604119583920, 4.31647844119580689503729587037, 4.75087494207642207202016603156, 5.50170953376952136648421110233, 6.09793983469196203141262840170, 6.42344517727849548057262253661, 7.56756149455693143394541194667
