L(s) = 1 | − 0.802·2-s + 3-s − 1.35·4-s + 0.612·5-s − 0.802·6-s − 2.13·7-s + 2.69·8-s + 9-s − 0.491·10-s + 2.38·11-s − 1.35·12-s + 3.73·13-s + 1.71·14-s + 0.612·15-s + 0.552·16-s + 17-s − 0.802·18-s − 0.128·19-s − 0.830·20-s − 2.13·21-s − 1.91·22-s − 8.18·23-s + 2.69·24-s − 4.62·25-s − 2.99·26-s + 27-s + 2.89·28-s + ⋯ |
L(s) = 1 | − 0.567·2-s + 0.577·3-s − 0.678·4-s + 0.273·5-s − 0.327·6-s − 0.806·7-s + 0.952·8-s + 0.333·9-s − 0.155·10-s + 0.719·11-s − 0.391·12-s + 1.03·13-s + 0.457·14-s + 0.158·15-s + 0.138·16-s + 0.242·17-s − 0.189·18-s − 0.0294·19-s − 0.185·20-s − 0.465·21-s − 0.407·22-s − 1.70·23-s + 0.549·24-s − 0.924·25-s − 0.587·26-s + 0.192·27-s + 0.547·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 + T \) |
good | 2 | \( 1 + 0.802T + 2T^{2} \) |
| 5 | \( 1 - 0.612T + 5T^{2} \) |
| 7 | \( 1 + 2.13T + 7T^{2} \) |
| 11 | \( 1 - 2.38T + 11T^{2} \) |
| 13 | \( 1 - 3.73T + 13T^{2} \) |
| 19 | \( 1 + 0.128T + 19T^{2} \) |
| 23 | \( 1 + 8.18T + 23T^{2} \) |
| 29 | \( 1 - 4.53T + 29T^{2} \) |
| 31 | \( 1 - 0.182T + 31T^{2} \) |
| 37 | \( 1 + 0.769T + 37T^{2} \) |
| 41 | \( 1 + 5.25T + 41T^{2} \) |
| 43 | \( 1 + 5.48T + 43T^{2} \) |
| 47 | \( 1 + 8.73T + 47T^{2} \) |
| 53 | \( 1 - 1.45T + 53T^{2} \) |
| 59 | \( 1 - 2.08T + 59T^{2} \) |
| 61 | \( 1 - 7.41T + 61T^{2} \) |
| 67 | \( 1 - 7.28T + 67T^{2} \) |
| 71 | \( 1 - 6.04T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 + 8.89T + 79T^{2} \) |
| 83 | \( 1 + 2.17T + 83T^{2} \) |
| 89 | \( 1 - 1.60T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−7.74589456671725612002287596330, −6.79114615341226885208170729502, −6.24745888211938110224944908164, −5.48758197449275333592106122129, −4.45723510687071228217989275662, −3.78950090206101298560208672532, −3.29186654734251964924726785921, −2.00969637387088447380061130970, −1.25585234214860008042344460713, 0,
1.25585234214860008042344460713, 2.00969637387088447380061130970, 3.29186654734251964924726785921, 3.78950090206101298560208672532, 4.45723510687071228217989275662, 5.48758197449275333592106122129, 6.24745888211938110224944908164, 6.79114615341226885208170729502, 7.74589456671725612002287596330