| L(s) = 1 | − 2-s − 2.23·3-s + 4-s + 0.618·5-s + 2.23·6-s + 7-s − 8-s + 2.00·9-s − 0.618·10-s − 2.23·12-s + 0.236·13-s − 14-s − 1.38·15-s + 16-s + 0.236·17-s − 2.00·18-s − 5·19-s + 0.618·20-s − 2.23·21-s − 0.236·23-s + 2.23·24-s − 4.61·25-s − 0.236·26-s + 2.23·27-s + 28-s + 1.47·29-s + 1.38·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.29·3-s + 0.5·4-s + 0.276·5-s + 0.912·6-s + 0.377·7-s − 0.353·8-s + 0.666·9-s − 0.195·10-s − 0.645·12-s + 0.0654·13-s − 0.267·14-s − 0.356·15-s + 0.250·16-s + 0.0572·17-s − 0.471·18-s − 1.14·19-s + 0.138·20-s − 0.487·21-s − 0.0492·23-s + 0.456·24-s − 0.923·25-s − 0.0462·26-s + 0.430·27-s + 0.188·28-s + 0.273·29-s + 0.252·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1694 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1694 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2.23T + 3T^{2} \) |
| 5 | \( 1 - 0.618T + 5T^{2} \) |
| 13 | \( 1 - 0.236T + 13T^{2} \) |
| 17 | \( 1 - 0.236T + 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + 0.236T + 23T^{2} \) |
| 29 | \( 1 - 1.47T + 29T^{2} \) |
| 31 | \( 1 - 8.70T + 31T^{2} \) |
| 37 | \( 1 + 7.85T + 37T^{2} \) |
| 41 | \( 1 + 8.94T + 41T^{2} \) |
| 43 | \( 1 - 1.85T + 43T^{2} \) |
| 47 | \( 1 - 6.61T + 47T^{2} \) |
| 53 | \( 1 - 8.23T + 53T^{2} \) |
| 59 | \( 1 - 8.61T + 59T^{2} \) |
| 61 | \( 1 + 5.32T + 61T^{2} \) |
| 67 | \( 1 - 9.18T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 + 9.61T + 73T^{2} \) |
| 79 | \( 1 + 15T + 79T^{2} \) |
| 83 | \( 1 - 8.61T + 83T^{2} \) |
| 89 | \( 1 + 2.90T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−8.827872302721467422843002791098, −8.322867496083470472526343261264, −7.23031851119677668344488353975, −6.49603249383870262147291990083, −5.83609049259298993837715424164, −5.04644024101135738881641669419, −4.04717606943887133184633192118, −2.52867450128108651652470189285, −1.34140465206346700767120531344, 0,
1.34140465206346700767120531344, 2.52867450128108651652470189285, 4.04717606943887133184633192118, 5.04644024101135738881641669419, 5.83609049259298993837715424164, 6.49603249383870262147291990083, 7.23031851119677668344488353975, 8.322867496083470472526343261264, 8.827872302721467422843002791098
