| L(s) = 1 | + 2-s + 3-s + 4-s + 4.36·5-s + 6-s + 4.52·7-s + 8-s + 9-s + 4.36·10-s + 11-s + 12-s − 6.17·13-s + 4.52·14-s + 4.36·15-s + 16-s + 4.36·17-s + 18-s − 5.40·19-s + 4.36·20-s + 4.52·21-s + 22-s − 7.65·23-s + 24-s + 14.0·25-s − 6.17·26-s + 27-s + 4.52·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.95·5-s + 0.408·6-s + 1.70·7-s + 0.353·8-s + 0.333·9-s + 1.38·10-s + 0.301·11-s + 0.288·12-s − 1.71·13-s + 1.20·14-s + 1.12·15-s + 0.250·16-s + 1.05·17-s + 0.235·18-s − 1.23·19-s + 0.977·20-s + 0.986·21-s + 0.213·22-s − 1.59·23-s + 0.204·24-s + 2.81·25-s − 1.21·26-s + 0.192·27-s + 0.854·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.266382562\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.266382562\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 5 | \( 1 - 4.36T + 5T^{2} \) |
| 7 | \( 1 - 4.52T + 7T^{2} \) |
| 13 | \( 1 + 6.17T + 13T^{2} \) |
| 17 | \( 1 - 4.36T + 17T^{2} \) |
| 19 | \( 1 + 5.40T + 19T^{2} \) |
| 23 | \( 1 + 7.65T + 23T^{2} \) |
| 29 | \( 1 + 7.78T + 29T^{2} \) |
| 31 | \( 1 + 1.28T + 31T^{2} \) |
| 37 | \( 1 + 0.611T + 37T^{2} \) |
| 41 | \( 1 - 9.12T + 41T^{2} \) |
| 43 | \( 1 + 9.01T + 43T^{2} \) |
| 47 | \( 1 + 0.764T + 47T^{2} \) |
| 53 | \( 1 + 0.479T + 53T^{2} \) |
| 59 | \( 1 - 7.96T + 59T^{2} \) |
| 67 | \( 1 + 6.16T + 67T^{2} \) |
| 71 | \( 1 - 8.06T + 71T^{2} \) |
| 73 | \( 1 + 6.83T + 73T^{2} \) |
| 79 | \( 1 - 2.99T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−8.360440952730460630674916011537, −7.68743845558300489657235041409, −6.95498778500302412242871978155, −5.96792006205516497523963831101, −5.41726746283321060898078072743, −4.80795139012144639589608708276, −4.01175268049694801052942102681, −2.62887761785164225856460799792, −2.02561692451421263865853388123, −1.57614262881653932199022393951,
1.57614262881653932199022393951, 2.02561692451421263865853388123, 2.62887761785164225856460799792, 4.01175268049694801052942102681, 4.80795139012144639589608708276, 5.41726746283321060898078072743, 5.96792006205516497523963831101, 6.95498778500302412242871978155, 7.68743845558300489657235041409, 8.360440952730460630674916011537
