| L(s) = 1 | + 2.08·2-s + 1.06·3-s + 2.35·4-s − 1.36·5-s + 2.23·6-s + 4.48·7-s + 0.733·8-s − 1.85·9-s − 2.84·10-s − 5.70·11-s + 2.51·12-s + 9.35·14-s − 1.45·15-s − 3.17·16-s − 4.60·17-s − 3.87·18-s + 2.65·19-s − 3.20·20-s + 4.79·21-s − 11.8·22-s − 6.30·23-s + 0.784·24-s − 3.14·25-s − 5.19·27-s + 10.5·28-s − 8.14·29-s − 3.03·30-s + ⋯ |
| L(s) = 1 | + 1.47·2-s + 0.617·3-s + 1.17·4-s − 0.609·5-s + 0.910·6-s + 1.69·7-s + 0.259·8-s − 0.618·9-s − 0.898·10-s − 1.71·11-s + 0.725·12-s + 2.50·14-s − 0.376·15-s − 0.793·16-s − 1.11·17-s − 0.912·18-s + 0.609·19-s − 0.716·20-s + 1.04·21-s − 2.53·22-s − 1.31·23-s + 0.160·24-s − 0.629·25-s − 0.999·27-s + 1.99·28-s − 1.51·29-s − 0.554·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 2.08T + 2T^{2} \) |
| 3 | \( 1 - 1.06T + 3T^{2} \) |
| 5 | \( 1 + 1.36T + 5T^{2} \) |
| 7 | \( 1 - 4.48T + 7T^{2} \) |
| 11 | \( 1 + 5.70T + 11T^{2} \) |
| 17 | \( 1 + 4.60T + 17T^{2} \) |
| 19 | \( 1 - 2.65T + 19T^{2} \) |
| 23 | \( 1 + 6.30T + 23T^{2} \) |
| 29 | \( 1 + 8.14T + 29T^{2} \) |
| 37 | \( 1 + 2.67T + 37T^{2} \) |
| 41 | \( 1 + 1.38T + 41T^{2} \) |
| 43 | \( 1 - 4.48T + 43T^{2} \) |
| 47 | \( 1 - 2.65T + 47T^{2} \) |
| 53 | \( 1 - 9.07T + 53T^{2} \) |
| 59 | \( 1 + 2.59T + 59T^{2} \) |
| 61 | \( 1 - 9.53T + 61T^{2} \) |
| 67 | \( 1 + 4.11T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 7.92T + 73T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 - 0.402T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−7.64825986994925859655979002313, −7.41133312930009012682094113369, −6.00676505070300949792971422937, −5.40570775063789901872442974208, −4.93443047699673375742916854432, −4.08887267067665684990223939535, −3.54794581436858311136225405330, −2.35085590606153154510542486721, −2.14368002761003622494947291246, 0,
2.14368002761003622494947291246, 2.35085590606153154510542486721, 3.54794581436858311136225405330, 4.08887267067665684990223939535, 4.93443047699673375742916854432, 5.40570775063789901872442974208, 6.00676505070300949792971422937, 7.41133312930009012682094113369, 7.64825986994925859655979002313
