| L(s) = 1 | + 3-s + 1.75·5-s − 2.58·7-s + 9-s + 11-s − 13-s + 1.75·15-s − 4.34·17-s + 4.79·19-s − 2.58·21-s − 5.83·23-s − 1.92·25-s + 27-s − 9.37·29-s + 7.14·31-s + 33-s − 4.54·35-s + 2.90·37-s − 39-s − 6.92·41-s − 0.133·43-s + 1.75·45-s + 9.59·47-s − 0.300·49-s − 4.34·51-s − 2·53-s + 1.75·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.784·5-s − 0.978·7-s + 0.333·9-s + 0.301·11-s − 0.277·13-s + 0.453·15-s − 1.05·17-s + 1.10·19-s − 0.564·21-s − 1.21·23-s − 0.384·25-s + 0.192·27-s − 1.74·29-s + 1.28·31-s + 0.174·33-s − 0.767·35-s + 0.478·37-s − 0.160·39-s − 1.08·41-s − 0.0203·43-s + 0.261·45-s + 1.39·47-s − 0.0428·49-s − 0.608·51-s − 0.274·53-s + 0.236·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 1.75T + 5T^{2} \) |
| 7 | \( 1 + 2.58T + 7T^{2} \) |
| 17 | \( 1 + 4.34T + 17T^{2} \) |
| 19 | \( 1 - 4.79T + 19T^{2} \) |
| 23 | \( 1 + 5.83T + 23T^{2} \) |
| 29 | \( 1 + 9.37T + 29T^{2} \) |
| 31 | \( 1 - 7.14T + 31T^{2} \) |
| 37 | \( 1 - 2.90T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 + 0.133T + 43T^{2} \) |
| 47 | \( 1 - 9.59T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 7.38T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 8.17T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 0.0978T + 73T^{2} \) |
| 79 | \( 1 + 6.34T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 0.945T + 89T^{2} \) |
| 97 | \( 1 - 2.90T + 97T^{2} \) |
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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
−7.50402337095595490495996610003, −6.99404400542938027369915804543, −6.08297825431626703958083338636, −5.77809062200774528885083415575, −4.64101113081238359297438878641, −3.89621258821204169436883507538, −3.08566134295612453109548105548, −2.32835233637072389998517566701, −1.51310498586409503511353922514, 0,
1.51310498586409503511353922514, 2.32835233637072389998517566701, 3.08566134295612453109548105548, 3.89621258821204169436883507538, 4.64101113081238359297438878641, 5.77809062200774528885083415575, 6.08297825431626703958083338636, 6.99404400542938027369915804543, 7.50402337095595490495996610003
