| L(s) = 1 | − 3-s + 1.28·5-s + 1.96·7-s + 9-s + 2.40·11-s − 3.92·13-s − 1.28·15-s − 3.63·17-s − 2.30·19-s − 1.96·21-s + 4.55·23-s − 3.34·25-s − 27-s − 5.81·29-s + 4.65·31-s − 2.40·33-s + 2.52·35-s + 4.09·37-s + 3.92·39-s + 7.34·41-s + 3.88·43-s + 1.28·45-s − 7.51·47-s − 3.14·49-s + 3.63·51-s − 10.6·53-s + 3.09·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.574·5-s + 0.742·7-s + 0.333·9-s + 0.726·11-s − 1.08·13-s − 0.331·15-s − 0.882·17-s − 0.529·19-s − 0.428·21-s + 0.948·23-s − 0.669·25-s − 0.192·27-s − 1.07·29-s + 0.836·31-s − 0.419·33-s + 0.426·35-s + 0.672·37-s + 0.628·39-s + 1.14·41-s + 0.592·43-s + 0.191·45-s − 1.09·47-s − 0.448·49-s + 0.509·51-s − 1.46·53-s + 0.417·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 1.28T + 5T^{2} \) |
| 7 | \( 1 - 1.96T + 7T^{2} \) |
| 11 | \( 1 - 2.40T + 11T^{2} \) |
| 13 | \( 1 + 3.92T + 13T^{2} \) |
| 17 | \( 1 + 3.63T + 17T^{2} \) |
| 19 | \( 1 + 2.30T + 19T^{2} \) |
| 23 | \( 1 - 4.55T + 23T^{2} \) |
| 29 | \( 1 + 5.81T + 29T^{2} \) |
| 31 | \( 1 - 4.65T + 31T^{2} \) |
| 37 | \( 1 - 4.09T + 37T^{2} \) |
| 41 | \( 1 - 7.34T + 41T^{2} \) |
| 43 | \( 1 - 3.88T + 43T^{2} \) |
| 47 | \( 1 + 7.51T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 + 0.286T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 3.44T + 73T^{2} \) |
| 79 | \( 1 + 5.99T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 + 9.05T + 89T^{2} \) |
| 97 | \( 1 - 4.02T + 97T^{2} \) |
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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
−7.51475384738666114380518170405, −6.62660260313448285929612940043, −6.18634457581730086337538820792, −5.37886622609778425519670735356, −4.62644160550012035525388958632, −4.25903198400247476224773692354, −2.97829058483177622653762337705, −2.06503311982854526750017771303, −1.35948939634690328495118280276, 0,
1.35948939634690328495118280276, 2.06503311982854526750017771303, 2.97829058483177622653762337705, 4.25903198400247476224773692354, 4.62644160550012035525388958632, 5.37886622609778425519670735356, 6.18634457581730086337538820792, 6.62660260313448285929612940043, 7.51475384738666114380518170405
