| L(s) = 1 | − 3-s − 3.40·5-s − 0.548·7-s + 9-s + 0.977·11-s − 1.94·13-s + 3.40·15-s + 1.38·17-s + 1.16·19-s + 0.548·21-s − 4.44·23-s + 6.60·25-s − 27-s + 3.71·29-s − 5.46·31-s − 0.977·33-s + 1.86·35-s − 2.93·37-s + 1.94·39-s + 6.35·41-s + 11.3·43-s − 3.40·45-s − 10.2·47-s − 6.69·49-s − 1.38·51-s + 3.23·53-s − 3.32·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.52·5-s − 0.207·7-s + 0.333·9-s + 0.294·11-s − 0.539·13-s + 0.879·15-s + 0.335·17-s + 0.266·19-s + 0.119·21-s − 0.926·23-s + 1.32·25-s − 0.192·27-s + 0.690·29-s − 0.981·31-s − 0.170·33-s + 0.315·35-s − 0.481·37-s + 0.311·39-s + 0.991·41-s + 1.72·43-s − 0.507·45-s − 1.49·47-s − 0.957·49-s − 0.193·51-s + 0.444·53-s − 0.448·55-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 | \( 1 + 3.40T + 5T^{2} \) |
| 7 | \( 1 + 0.548T + 7T^{2} \) |
| 11 | \( 1 - 0.977T + 11T^{2} \) |
| 13 | \( 1 + 1.94T + 13T^{2} \) |
| 17 | \( 1 - 1.38T + 17T^{2} \) |
| 19 | \( 1 - 1.16T + 19T^{2} \) |
| 23 | \( 1 + 4.44T + 23T^{2} \) |
| 29 | \( 1 - 3.71T + 29T^{2} \) |
| 31 | \( 1 + 5.46T + 31T^{2} \) |
| 37 | \( 1 + 2.93T + 37T^{2} \) |
| 41 | \( 1 - 6.35T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 3.23T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 + 14.9T + 61T^{2} \) |
| 67 | \( 1 - 15.2T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 0.389T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 - 5.41T + 83T^{2} \) |
| 89 | \( 1 + 17.1T + 89T^{2} \) |
| 97 | \( 1 + 2.62T + 97T^{2} \) |
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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
−7.54811406609310096882686791781, −6.87419877933558331391529887776, −6.17773936662640608705013476252, −5.31326934584837806657363012988, −4.61558580208166940505317508579, −3.90012637253628638187965128118, −3.37061730864962051933823796506, −2.24670329848124388112990588381, −0.937421823108086867613207509220, 0,
0.937421823108086867613207509220, 2.24670329848124388112990588381, 3.37061730864962051933823796506, 3.90012637253628638187965128118, 4.61558580208166940505317508579, 5.31326934584837806657363012988, 6.17773936662640608705013476252, 6.87419877933558331391529887776, 7.54811406609310096882686791781
