L(s) = 1 | + 2-s + 3-s + 4-s − 0.141·5-s + 6-s − 4.12·7-s + 8-s + 9-s − 0.141·10-s − 11-s + 12-s + 3.20·13-s − 4.12·14-s − 0.141·15-s + 16-s − 6.07·17-s + 18-s − 2.09·19-s − 0.141·20-s − 4.12·21-s − 22-s + 1.03·23-s + 24-s − 4.97·25-s + 3.20·26-s + 27-s − 4.12·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.0633·5-s + 0.408·6-s − 1.55·7-s + 0.353·8-s + 0.333·9-s − 0.0447·10-s − 0.301·11-s + 0.288·12-s + 0.889·13-s − 1.10·14-s − 0.0365·15-s + 0.250·16-s − 1.47·17-s + 0.235·18-s − 0.480·19-s − 0.0316·20-s − 0.899·21-s − 0.213·22-s + 0.216·23-s + 0.204·24-s − 0.995·25-s + 0.628·26-s + 0.192·27-s − 0.778·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
good | 5 | \( 1 + 0.141T + 5T^{2} \) |
| 7 | \( 1 + 4.12T + 7T^{2} \) |
| 13 | \( 1 - 3.20T + 13T^{2} \) |
| 17 | \( 1 + 6.07T + 17T^{2} \) |
| 19 | \( 1 + 2.09T + 19T^{2} \) |
| 23 | \( 1 - 1.03T + 23T^{2} \) |
| 29 | \( 1 - 3.48T + 29T^{2} \) |
| 31 | \( 1 - 8.30T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 + 3.63T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 + 6.13T + 59T^{2} \) |
| 67 | \( 1 - 4.89T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 - 3.28T + 79T^{2} \) |
| 83 | \( 1 - 8.23T + 83T^{2} \) |
| 89 | \( 1 - 6.57T + 89T^{2} \) |
| 97 | \( 1 - 1.38T + 97T^{2} \) |
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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
−8.293033544579553101860727367246, −6.90650247153212910611808108176, −6.72342181508508752358226241561, −5.99568862329156442200308120928, −4.96613608641697145730008799958, −4.08163457892087109561411239011, −3.40652406369662977764504013180, −2.76035917629323814000127050971, −1.74879753788453997010423519259, 0,
1.74879753788453997010423519259, 2.76035917629323814000127050971, 3.40652406369662977764504013180, 4.08163457892087109561411239011, 4.96613608641697145730008799958, 5.99568862329156442200308120928, 6.72342181508508752358226241561, 6.90650247153212910611808108176, 8.293033544579553101860727367246