| L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s − 11-s + 5.46·13-s + 14-s + 16-s + 3.46·17-s + 0.732·19-s − 20-s − 22-s − 4.73·23-s + 25-s + 5.46·26-s + 28-s + 1.26·29-s − 4.92·31-s + 32-s + 3.46·34-s − 35-s + 6.73·37-s + 0.732·38-s − 40-s + 1.26·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.301·11-s + 1.51·13-s + 0.267·14-s + 0.250·16-s + 0.840·17-s + 0.167·19-s − 0.223·20-s − 0.213·22-s − 0.986·23-s + 0.200·25-s + 1.07·26-s + 0.188·28-s + 0.235·29-s − 0.885·31-s + 0.176·32-s + 0.594·34-s − 0.169·35-s + 1.10·37-s + 0.118·38-s − 0.158·40-s + 0.198·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.476008051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.476008051\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 5.46T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 0.732T + 19T^{2} \) |
| 23 | \( 1 + 4.73T + 23T^{2} \) |
| 29 | \( 1 - 1.26T + 29T^{2} \) |
| 31 | \( 1 + 4.92T + 31T^{2} \) |
| 37 | \( 1 - 6.73T + 37T^{2} \) |
| 41 | \( 1 - 1.26T + 41T^{2} \) |
| 43 | \( 1 - 8.92T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 1.26T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 2.92T + 67T^{2} \) |
| 71 | \( 1 + 2.53T + 71T^{2} \) |
| 73 | \( 1 - 4.53T + 73T^{2} \) |
| 79 | \( 1 - 3.26T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 - 8.53T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−7.75655587775575908271257549236, −7.43142441911918181066357031300, −6.24907879193331714577530983211, −5.92961118492704719945255369372, −5.10481920482527062188811603656, −4.24853701323207272660718394511, −3.70267479086179153307686302042, −2.94173907749004183753286302806, −1.87572533000356632405822892559, −0.889705387371480721795198427193,
0.889705387371480721795198427193, 1.87572533000356632405822892559, 2.94173907749004183753286302806, 3.70267479086179153307686302042, 4.24853701323207272660718394511, 5.10481920482527062188811603656, 5.92961118492704719945255369372, 6.24907879193331714577530983211, 7.43142441911918181066357031300, 7.75655587775575908271257549236
