| L(s) = 1 | + 3.18·3-s − 1.57·5-s − 3.06·7-s + 7.15·9-s + 0.545·11-s − 1.64·13-s − 5.00·15-s − 3.65·17-s − 2.61·19-s − 9.75·21-s − 3.92·23-s − 2.53·25-s + 13.2·27-s + 7.11·29-s − 7.75·31-s + 1.73·33-s + 4.80·35-s − 9.57·37-s − 5.24·39-s − 1.04·41-s + 10.0·43-s − 11.2·45-s − 2.76·47-s + 2.36·49-s − 11.6·51-s − 9.38·53-s − 0.856·55-s + ⋯ |
| L(s) = 1 | + 1.83·3-s − 0.702·5-s − 1.15·7-s + 2.38·9-s + 0.164·11-s − 0.456·13-s − 1.29·15-s − 0.885·17-s − 0.599·19-s − 2.12·21-s − 0.819·23-s − 0.506·25-s + 2.54·27-s + 1.32·29-s − 1.39·31-s + 0.302·33-s + 0.812·35-s − 1.57·37-s − 0.840·39-s − 0.162·41-s + 1.53·43-s − 1.67·45-s − 0.402·47-s + 0.338·49-s − 1.62·51-s − 1.28·53-s − 0.115·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - 3.18T + 3T^{2} \) |
| 5 | \( 1 + 1.57T + 5T^{2} \) |
| 7 | \( 1 + 3.06T + 7T^{2} \) |
| 11 | \( 1 - 0.545T + 11T^{2} \) |
| 13 | \( 1 + 1.64T + 13T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 + 2.61T + 19T^{2} \) |
| 23 | \( 1 + 3.92T + 23T^{2} \) |
| 29 | \( 1 - 7.11T + 29T^{2} \) |
| 31 | \( 1 + 7.75T + 31T^{2} \) |
| 37 | \( 1 + 9.57T + 37T^{2} \) |
| 41 | \( 1 + 1.04T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 + 2.76T + 47T^{2} \) |
| 53 | \( 1 + 9.38T + 53T^{2} \) |
| 59 | \( 1 + 2.80T + 59T^{2} \) |
| 61 | \( 1 - 5.26T + 61T^{2} \) |
| 67 | \( 1 - 7.28T + 67T^{2} \) |
| 71 | \( 1 + 3.46T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 + 9.12T + 79T^{2} \) |
| 83 | \( 1 - 5.33T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−8.196909209972682371221166816898, −7.42359083609211649095090203102, −6.92001723445394775643408112081, −6.10165623376747825076224727123, −4.66717816497136907065051284242, −3.95794801779778709828956018095, −3.40801113409661280738840292149, −2.61943661380154632122458572000, −1.80520640790212388090874593432, 0,
1.80520640790212388090874593432, 2.61943661380154632122458572000, 3.40801113409661280738840292149, 3.95794801779778709828956018095, 4.66717816497136907065051284242, 6.10165623376747825076224727123, 6.92001723445394775643408112081, 7.42359083609211649095090203102, 8.196909209972682371221166816898
