L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 3·7-s − 8-s + 9-s − 10-s + 0.267·11-s − 12-s + 3·14-s − 15-s + 16-s − 4·17-s − 18-s + 5.73·19-s + 20-s + 3·21-s − 0.267·22-s − 3.46·23-s + 24-s + 25-s − 27-s − 3·28-s + 1.46·29-s + 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.0807·11-s − 0.288·12-s + 0.801·14-s − 0.258·15-s + 0.250·16-s − 0.970·17-s − 0.235·18-s + 1.31·19-s + 0.223·20-s + 0.654·21-s − 0.0571·22-s − 0.722·23-s + 0.204·24-s + 0.200·25-s − 0.192·27-s − 0.566·28-s + 0.271·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 0.267T + 11T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 5.73T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 1.46T + 29T^{2} \) |
| 31 | \( 1 - 4.92T + 31T^{2} \) |
| 37 | \( 1 + 5.92T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 - 6.46T + 47T^{2} \) |
| 53 | \( 1 + 0.267T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 0.535T + 61T^{2} \) |
| 67 | \( 1 + 1.46T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 - 3.07T + 79T^{2} \) |
| 83 | \( 1 - 9.46T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 - 8.39T + 97T^{2} \) |
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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
−7.86046389514128404549367068869, −6.96939823295719878145452420785, −6.57695750305345689497490233316, −5.87012850332408645937059775563, −5.15795260887058384132360691115, −4.08549948378650356597657732780, −3.14562861509133298524521506342, −2.28340704680664953639672420367, −1.13064255016672803054918077780, 0,
1.13064255016672803054918077780, 2.28340704680664953639672420367, 3.14562861509133298524521506342, 4.08549948378650356597657732780, 5.15795260887058384132360691115, 5.87012850332408645937059775563, 6.57695750305345689497490233316, 6.96939823295719878145452420785, 7.86046389514128404549367068869