L(s) = 1 | − 2.14·2-s + 3-s + 2.59·4-s + 0.772·5-s − 2.14·6-s − 7-s − 1.27·8-s + 9-s − 1.65·10-s + 3.96·11-s + 2.59·12-s + 5.13·13-s + 2.14·14-s + 0.772·15-s − 2.45·16-s − 7.18·17-s − 2.14·18-s + 2.38·19-s + 2.00·20-s − 21-s − 8.49·22-s − 7.93·23-s − 1.27·24-s − 4.40·25-s − 11.0·26-s + 27-s − 2.59·28-s + ⋯ |
L(s) = 1 | − 1.51·2-s + 0.577·3-s + 1.29·4-s + 0.345·5-s − 0.875·6-s − 0.377·7-s − 0.450·8-s + 0.333·9-s − 0.523·10-s + 1.19·11-s + 0.749·12-s + 1.42·13-s + 0.572·14-s + 0.199·15-s − 0.614·16-s − 1.74·17-s − 0.505·18-s + 0.546·19-s + 0.447·20-s − 0.218·21-s − 1.81·22-s − 1.65·23-s − 0.260·24-s − 0.880·25-s − 2.15·26-s + 0.192·27-s − 0.490·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 383 | \( 1 - T \) |
good | 2 | \( 1 + 2.14T + 2T^{2} \) |
| 5 | \( 1 - 0.772T + 5T^{2} \) |
| 11 | \( 1 - 3.96T + 11T^{2} \) |
| 13 | \( 1 - 5.13T + 13T^{2} \) |
| 17 | \( 1 + 7.18T + 17T^{2} \) |
| 19 | \( 1 - 2.38T + 19T^{2} \) |
| 23 | \( 1 + 7.93T + 23T^{2} \) |
| 29 | \( 1 - 2.67T + 29T^{2} \) |
| 31 | \( 1 - 0.795T + 31T^{2} \) |
| 37 | \( 1 + 4.04T + 37T^{2} \) |
| 41 | \( 1 + 6.57T + 41T^{2} \) |
| 43 | \( 1 - 4.07T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 - 1.03T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 + 6.46T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 6.22T + 71T^{2} \) |
| 73 | \( 1 - 1.99T + 73T^{2} \) |
| 79 | \( 1 + 9.48T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 + 6.26T + 89T^{2} \) |
| 97 | \( 1 + 9.89T + 97T^{2} \) |
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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
−7.72524378468260017459062937696, −6.88566951941586714466198703873, −6.41352900645005316002485075827, −5.80099268327709246579127321321, −4.30072527979013917033073389261, −3.93427596250463033250413176249, −2.79597765020280293081608578533, −1.84433728051519338240491949226, −1.33266286424725653738680777806, 0,
1.33266286424725653738680777806, 1.84433728051519338240491949226, 2.79597765020280293081608578533, 3.93427596250463033250413176249, 4.30072527979013917033073389261, 5.80099268327709246579127321321, 6.41352900645005316002485075827, 6.88566951941586714466198703873, 7.72524378468260017459062937696