| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s + 16.3·5-s − 6·6-s + 8.04·7-s + 8·8-s + 9·9-s + 32.7·10-s − 31.3·11-s − 12·12-s − 8.08·13-s + 16.0·14-s − 49.1·15-s + 16·16-s + 18·18-s − 106.·19-s + 65.5·20-s − 24.1·21-s − 62.7·22-s − 169.·23-s − 24·24-s + 143.·25-s − 16.1·26-s − 27·27-s + 32.1·28-s − 9.93·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.46·5-s − 0.408·6-s + 0.434·7-s + 0.353·8-s + 0.333·9-s + 1.03·10-s − 0.859·11-s − 0.288·12-s − 0.172·13-s + 0.307·14-s − 0.846·15-s + 0.250·16-s + 0.235·18-s − 1.28·19-s + 0.733·20-s − 0.250·21-s − 0.608·22-s − 1.53·23-s − 0.204·24-s + 1.15·25-s − 0.121·26-s − 0.192·27-s + 0.217·28-s − 0.0636·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 16.3T + 125T^{2} \) |
| 7 | \( 1 - 8.04T + 343T^{2} \) |
| 11 | \( 1 + 31.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 8.08T + 2.19e3T^{2} \) |
| 19 | \( 1 + 106.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 169.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 9.93T + 2.43e4T^{2} \) |
| 31 | \( 1 + 300.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 176.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 135.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 467.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 491.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 387.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 137.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 289.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 52.9T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.05e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 133.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 214.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 438.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 561.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.30e3T + 9.12e5T^{2} \) |
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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
−8.472311942626855601980625681373, −7.67108581866052822397266186343, −6.52032841536281220444452548934, −6.10414858982690356604737747916, −5.24466045562250212386225913280, −4.75863958671843700939517731099, −3.52868682978853037212423394725, −2.15721843611367683405314813463, −1.77405402366620688952273844163, 0,
1.77405402366620688952273844163, 2.15721843611367683405314813463, 3.52868682978853037212423394725, 4.75863958671843700939517731099, 5.24466045562250212386225913280, 6.10414858982690356604737747916, 6.52032841536281220444452548934, 7.67108581866052822397266186343, 8.472311942626855601980625681373
