| L(s) = 1 | + 1.80·2-s − 3-s + 1.24·4-s − 3.07·5-s − 1.80·6-s + 7-s − 1.35·8-s + 9-s − 5.54·10-s + 6.47·11-s − 1.24·12-s + 1.80·14-s + 3.07·15-s − 4.93·16-s − 2.47·17-s + 1.80·18-s − 5.83·19-s − 3.83·20-s − 21-s + 11.6·22-s + 3.64·23-s + 1.35·24-s + 4.47·25-s − 27-s + 1.24·28-s + 4.48·29-s + 5.54·30-s + ⋯ |
| L(s) = 1 | + 1.27·2-s − 0.577·3-s + 0.623·4-s − 1.37·5-s − 0.735·6-s + 0.377·7-s − 0.479·8-s + 0.333·9-s − 1.75·10-s + 1.95·11-s − 0.359·12-s + 0.481·14-s + 0.794·15-s − 1.23·16-s − 0.601·17-s + 0.424·18-s − 1.33·19-s − 0.858·20-s − 0.218·21-s + 2.48·22-s + 0.759·23-s + 0.276·24-s + 0.894·25-s − 0.192·27-s + 0.235·28-s + 0.833·29-s + 1.01·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.80T + 2T^{2} \) |
| 5 | \( 1 + 3.07T + 5T^{2} \) |
| 11 | \( 1 - 6.47T + 11T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 19 | \( 1 + 5.83T + 19T^{2} \) |
| 23 | \( 1 - 3.64T + 23T^{2} \) |
| 29 | \( 1 - 4.48T + 29T^{2} \) |
| 31 | \( 1 - 2.78T + 31T^{2} \) |
| 37 | \( 1 + 1.16T + 37T^{2} \) |
| 41 | \( 1 + 1.53T + 41T^{2} \) |
| 43 | \( 1 + 6.89T + 43T^{2} \) |
| 47 | \( 1 + 5.15T + 47T^{2} \) |
| 53 | \( 1 + 9.65T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 + 7.28T + 61T^{2} \) |
| 67 | \( 1 - 2.56T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 - 7.58T + 83T^{2} \) |
| 89 | \( 1 + 18.6T + 89T^{2} \) |
| 97 | \( 1 + 1.08T + 97T^{2} \) |
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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
−8.289257194264088205858187236024, −6.92296303783215679767175921402, −6.73226139102957091071149944000, −5.91460100904676913759669778194, −4.78183230950898559173747312475, −4.36147147650890042012552587113, −3.87448091125198484682948910216, −2.96682209948350230631487459743, −1.47566443882539701186888727303, 0,
1.47566443882539701186888727303, 2.96682209948350230631487459743, 3.87448091125198484682948910216, 4.36147147650890042012552587113, 4.78183230950898559173747312475, 5.91460100904676913759669778194, 6.73226139102957091071149944000, 6.92296303783215679767175921402, 8.289257194264088205858187236024
