| L(s) = 1 | − 2-s − 2.77·3-s + 4-s − 0.764·5-s + 2.77·6-s + 0.951·7-s − 8-s + 4.72·9-s + 0.764·10-s + 4.41·11-s − 2.77·12-s − 0.951·14-s + 2.12·15-s + 16-s − 3.70·17-s − 4.72·18-s − 19-s − 0.764·20-s − 2.64·21-s − 4.41·22-s + 9.20·23-s + 2.77·24-s − 4.41·25-s − 4.80·27-s + 0.951·28-s − 1.83·29-s − 2.12·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.60·3-s + 0.5·4-s − 0.342·5-s + 1.13·6-s + 0.359·7-s − 0.353·8-s + 1.57·9-s + 0.241·10-s + 1.33·11-s − 0.802·12-s − 0.254·14-s + 0.548·15-s + 0.250·16-s − 0.897·17-s − 1.11·18-s − 0.229·19-s − 0.171·20-s − 0.577·21-s − 0.940·22-s + 1.91·23-s + 0.567·24-s − 0.883·25-s − 0.924·27-s + 0.179·28-s − 0.340·29-s − 0.388·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8103782616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8103782616\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.77T + 3T^{2} \) |
| 5 | \( 1 + 0.764T + 5T^{2} \) |
| 7 | \( 1 - 0.951T + 7T^{2} \) |
| 11 | \( 1 - 4.41T + 11T^{2} \) |
| 17 | \( 1 + 3.70T + 17T^{2} \) |
| 23 | \( 1 - 9.20T + 23T^{2} \) |
| 29 | \( 1 + 1.83T + 29T^{2} \) |
| 31 | \( 1 - 4.09T + 31T^{2} \) |
| 37 | \( 1 - 4.97T + 37T^{2} \) |
| 41 | \( 1 - 2.26T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 - 7.33T + 47T^{2} \) |
| 53 | \( 1 - 2.67T + 53T^{2} \) |
| 59 | \( 1 + 5.46T + 59T^{2} \) |
| 61 | \( 1 - 5.24T + 61T^{2} \) |
| 67 | \( 1 - 8.86T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 - 9.56T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 - 5.49T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−7.891194572804605234313136396150, −7.18577621013797896942869910204, −6.58326950055047187095892555864, −6.11277724728427145137902691541, −5.27770365557549602867609879932, −4.48201909889612210977988674797, −3.87808062097050587795403013759, −2.51080292915790018628495814757, −1.32477879861516692624576058346, −0.64459711574933456383513639045,
0.64459711574933456383513639045, 1.32477879861516692624576058346, 2.51080292915790018628495814757, 3.87808062097050587795403013759, 4.48201909889612210977988674797, 5.27770365557549602867609879932, 6.11277724728427145137902691541, 6.58326950055047187095892555864, 7.18577621013797896942869910204, 7.891194572804605234313136396150
