| L(s) = 1 | − 3-s + 4.36·5-s − 0.777·7-s + 9-s − 11-s − 13-s − 4.36·15-s − 5.14·17-s − 0.559·19-s + 0.777·21-s + 0.777·23-s + 14.0·25-s − 27-s − 3.80·29-s − 10.5·31-s + 33-s − 3.39·35-s − 8.28·37-s + 39-s + 10.3·41-s + 4.48·43-s + 4.36·45-s − 8·47-s − 6.39·49-s + 5.14·51-s + 6·53-s − 4.36·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.95·5-s − 0.293·7-s + 0.333·9-s − 0.301·11-s − 0.277·13-s − 1.12·15-s − 1.24·17-s − 0.128·19-s + 0.169·21-s + 0.162·23-s + 2.81·25-s − 0.192·27-s − 0.706·29-s − 1.90·31-s + 0.174·33-s − 0.573·35-s − 1.36·37-s + 0.160·39-s + 1.62·41-s + 0.683·43-s + 0.650·45-s − 1.16·47-s − 0.913·49-s + 0.720·51-s + 0.824·53-s − 0.588·55-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 | \( 1 - 4.36T + 5T^{2} \) |
| 7 | \( 1 + 0.777T + 7T^{2} \) |
| 17 | \( 1 + 5.14T + 17T^{2} \) |
| 19 | \( 1 + 0.559T + 19T^{2} \) |
| 23 | \( 1 - 0.777T + 23T^{2} \) |
| 29 | \( 1 + 3.80T + 29T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 37 | \( 1 + 8.28T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 4.48T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 0.217T + 59T^{2} \) |
| 61 | \( 1 - 2.21T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 + 0.855T + 79T^{2} \) |
| 83 | \( 1 + 6.67T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 8.28T + 97T^{2} \) |
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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
−7.28276470934282663196315429034, −6.82924220318547015386670833437, −6.08400219646387163626884490770, −5.57547273754440958769841486361, −5.03906702933994279310281610620, −4.13231054167291820952490080419, −2.93595361003841850332637186963, −2.13394164846692831386730514564, −1.49502691646600597848655233106, 0,
1.49502691646600597848655233106, 2.13394164846692831386730514564, 2.93595361003841850332637186963, 4.13231054167291820952490080419, 5.03906702933994279310281610620, 5.57547273754440958769841486361, 6.08400219646387163626884490770, 6.82924220318547015386670833437, 7.28276470934282663196315429034
