| L(s) = 1 | − 2-s + 3-s + 4-s − 2.59·5-s − 6-s + 7-s − 8-s + 9-s + 2.59·10-s − 5.12·11-s + 12-s − 14-s − 2.59·15-s + 16-s − 3.39·17-s − 18-s − 0.0178·19-s − 2.59·20-s + 21-s + 5.12·22-s + 7.09·23-s − 24-s + 1.73·25-s + 27-s + 28-s + 4.27·29-s + 2.59·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.16·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.820·10-s − 1.54·11-s + 0.288·12-s − 0.267·14-s − 0.670·15-s + 0.250·16-s − 0.823·17-s − 0.235·18-s − 0.00409·19-s − 0.580·20-s + 0.218·21-s + 1.09·22-s + 1.47·23-s − 0.204·24-s + 0.347·25-s + 0.192·27-s + 0.188·28-s + 0.794·29-s + 0.473·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2.59T + 5T^{2} \) |
| 11 | \( 1 + 5.12T + 11T^{2} \) |
| 17 | \( 1 + 3.39T + 17T^{2} \) |
| 19 | \( 1 + 0.0178T + 19T^{2} \) |
| 23 | \( 1 - 7.09T + 23T^{2} \) |
| 29 | \( 1 - 4.27T + 29T^{2} \) |
| 31 | \( 1 - 0.430T + 31T^{2} \) |
| 37 | \( 1 - 4.46T + 37T^{2} \) |
| 41 | \( 1 + 2.69T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 5.56T + 47T^{2} \) |
| 53 | \( 1 + 9.44T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 + 7.25T + 61T^{2} \) |
| 67 | \( 1 - 1.37T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 - 3.14T + 73T^{2} \) |
| 79 | \( 1 - 7.29T + 79T^{2} \) |
| 83 | \( 1 + 2.41T + 83T^{2} \) |
| 89 | \( 1 - 2.71T + 89T^{2} \) |
| 97 | \( 1 + 16.7T + 97T^{2} \) |
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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
−7.68600161221350200453634892014, −7.28727650107300604772123132593, −6.46872393890687779823689766480, −5.37627931902118967979090115002, −4.65206528672549789306984533367, −3.91938502754205827444601850925, −2.85849412696442557493576407166, −2.46709286479969133417898342283, −1.11543143655558947155396062939, 0,
1.11543143655558947155396062939, 2.46709286479969133417898342283, 2.85849412696442557493576407166, 3.91938502754205827444601850925, 4.65206528672549789306984533367, 5.37627931902118967979090115002, 6.46872393890687779823689766480, 7.28727650107300604772123132593, 7.68600161221350200453634892014
