L(s) = 1 | − 3-s + 3.46·5-s + 7-s + 9-s − 2·13-s − 3.46·15-s − 3.46·17-s + 19-s − 21-s − 4·23-s + 6.99·25-s − 27-s − 0.535·29-s − 10.9·31-s + 3.46·35-s − 2·37-s + 2·39-s + 6·41-s − 6.92·43-s + 3.46·45-s − 9.46·47-s + 49-s + 3.46·51-s − 3.46·53-s − 57-s − 4·59-s + 6·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.54·5-s + 0.377·7-s + 0.333·9-s − 0.554·13-s − 0.894·15-s − 0.840·17-s + 0.229·19-s − 0.218·21-s − 0.834·23-s + 1.39·25-s − 0.192·27-s − 0.0995·29-s − 1.96·31-s + 0.585·35-s − 0.328·37-s + 0.320·39-s + 0.937·41-s − 1.05·43-s + 0.516·45-s − 1.38·47-s + 0.142·49-s + 0.485·51-s − 0.475·53-s − 0.132·57-s − 0.520·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 - 3.46T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 0.535T + 29T^{2} \) |
| 31 | \( 1 + 10.9T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 6.92T + 43T^{2} \) |
| 47 | \( 1 + 9.46T + 47T^{2} \) |
| 53 | \( 1 + 3.46T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 + 0.928T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 5.46T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−7.50784955470981668478087344720, −6.87083343128890967239573089003, −6.11666235978260315106507213083, −5.60912840937199909173260299070, −4.97815822090921893042469726894, −4.23777221874247496611362479275, −3.05654771172371249414613878305, −2.00978967669075797575224498209, −1.58207393914514526105664174470, 0,
1.58207393914514526105664174470, 2.00978967669075797575224498209, 3.05654771172371249414613878305, 4.23777221874247496611362479275, 4.97815822090921893042469726894, 5.60912840937199909173260299070, 6.11666235978260315106507213083, 6.87083343128890967239573089003, 7.50784955470981668478087344720