| L(s) = 1 | − 3-s + 3.06·5-s + 2.07·7-s + 9-s − 4.83·11-s + 4.75·13-s − 3.06·15-s − 6.72·17-s − 7.15·19-s − 2.07·21-s + 1.61·23-s + 4.41·25-s − 27-s − 3.03·29-s + 6.60·31-s + 4.83·33-s + 6.36·35-s − 8.82·37-s − 4.75·39-s + 5.00·41-s + 2.85·43-s + 3.06·45-s + 9.49·47-s − 2.69·49-s + 6.72·51-s − 2.12·53-s − 14.8·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.37·5-s + 0.784·7-s + 0.333·9-s − 1.45·11-s + 1.31·13-s − 0.792·15-s − 1.62·17-s − 1.64·19-s − 0.452·21-s + 0.335·23-s + 0.882·25-s − 0.192·27-s − 0.563·29-s + 1.18·31-s + 0.842·33-s + 1.07·35-s − 1.45·37-s − 0.761·39-s + 0.781·41-s + 0.435·43-s + 0.457·45-s + 1.38·47-s − 0.384·49-s + 0.941·51-s − 0.291·53-s − 2.00·55-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 | \( 1 - 3.06T + 5T^{2} \) |
| 7 | \( 1 - 2.07T + 7T^{2} \) |
| 11 | \( 1 + 4.83T + 11T^{2} \) |
| 13 | \( 1 - 4.75T + 13T^{2} \) |
| 17 | \( 1 + 6.72T + 17T^{2} \) |
| 19 | \( 1 + 7.15T + 19T^{2} \) |
| 23 | \( 1 - 1.61T + 23T^{2} \) |
| 29 | \( 1 + 3.03T + 29T^{2} \) |
| 31 | \( 1 - 6.60T + 31T^{2} \) |
| 37 | \( 1 + 8.82T + 37T^{2} \) |
| 41 | \( 1 - 5.00T + 41T^{2} \) |
| 43 | \( 1 - 2.85T + 43T^{2} \) |
| 47 | \( 1 - 9.49T + 47T^{2} \) |
| 53 | \( 1 + 2.12T + 53T^{2} \) |
| 59 | \( 1 + 8.57T + 59T^{2} \) |
| 61 | \( 1 + 7.09T + 61T^{2} \) |
| 67 | \( 1 + 7.12T + 67T^{2} \) |
| 71 | \( 1 + 1.95T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 - 1.94T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 + 16.6T + 97T^{2} \) |
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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
−7.41387277150100427583854987583, −6.56154868839185117145403123358, −6.03487070076443156424776782198, −5.55145988807934084107040086768, −4.69811275353630737494824600411, −4.24388872817135736249405470453, −2.82703335523606538864917108065, −2.09895449876566552104931018709, −1.44625540158056021664896258338, 0,
1.44625540158056021664896258338, 2.09895449876566552104931018709, 2.82703335523606538864917108065, 4.24388872817135736249405470453, 4.69811275353630737494824600411, 5.55145988807934084107040086768, 6.03487070076443156424776782198, 6.56154868839185117145403123358, 7.41387277150100427583854987583
