| L(s) = 1 | − 2-s + 6·3-s − 7·4-s − 6·6-s + 15·8-s + 9·9-s − 44·11-s − 42·12-s + 6·13-s + 41·16-s − 24·17-s − 9·18-s + 114·19-s + 44·22-s + 52·23-s + 90·24-s − 6·26-s − 108·27-s + 146·29-s + 276·31-s − 161·32-s − 264·33-s + 24·34-s − 63·36-s + 210·37-s − 114·38-s + 36·39-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 1.15·3-s − 7/8·4-s − 0.408·6-s + 0.662·8-s + 1/3·9-s − 1.20·11-s − 1.01·12-s + 0.128·13-s + 0.640·16-s − 0.342·17-s − 0.117·18-s + 1.37·19-s + 0.426·22-s + 0.471·23-s + 0.765·24-s − 0.0452·26-s − 0.769·27-s + 0.934·29-s + 1.59·31-s − 0.889·32-s − 1.39·33-s + 0.121·34-s − 0.291·36-s + 0.933·37-s − 0.486·38-s + 0.147·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 3 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 - 6 T + p^{3} T^{2} \) |
| 17 | \( 1 + 24 T + p^{3} T^{2} \) |
| 19 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 23 | \( 1 - 52 T + p^{3} T^{2} \) |
| 29 | \( 1 - 146 T + p^{3} T^{2} \) |
| 31 | \( 1 - 276 T + p^{3} T^{2} \) |
| 37 | \( 1 - 210 T + p^{3} T^{2} \) |
| 41 | \( 1 + 444 T + p^{3} T^{2} \) |
| 43 | \( 1 + 492 T + p^{3} T^{2} \) |
| 47 | \( 1 + 612 T + p^{3} T^{2} \) |
| 53 | \( 1 + 50 T + p^{3} T^{2} \) |
| 59 | \( 1 + 294 T + p^{3} T^{2} \) |
| 61 | \( 1 + 450 T + p^{3} T^{2} \) |
| 67 | \( 1 - 668 T + p^{3} T^{2} \) |
| 71 | \( 1 + 308 T + p^{3} T^{2} \) |
| 73 | \( 1 - 12 T + p^{3} T^{2} \) |
| 79 | \( 1 - 596 T + p^{3} T^{2} \) |
| 83 | \( 1 + 966 T + p^{3} T^{2} \) |
| 89 | \( 1 - 408 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1200 T + p^{3} T^{2} \) |
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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
−8.803547610575346675737094727475, −8.165300067815412051158050608127, −7.82203220619479855097087255746, −6.63783921355095265028737520511, −5.27493328830885711235376191401, −4.67194896081682199760347337220, −3.38661960132530435021790383896, −2.77445787987338686290477665031, −1.37360245590215893343146589244, 0,
1.37360245590215893343146589244, 2.77445787987338686290477665031, 3.38661960132530435021790383896, 4.67194896081682199760347337220, 5.27493328830885711235376191401, 6.63783921355095265028737520511, 7.82203220619479855097087255746, 8.165300067815412051158050608127, 8.803547610575346675737094727475
