| L(s) = 1 | + 2-s − 3-s + 4-s + 3.72·5-s − 6-s − 0.968·7-s + 8-s + 9-s + 3.72·10-s + 3.53·11-s − 12-s − 4.84·13-s − 0.968·14-s − 3.72·15-s + 16-s + 8.09·17-s + 18-s − 5.04·19-s + 3.72·20-s + 0.968·21-s + 3.53·22-s − 4.28·23-s − 24-s + 8.87·25-s − 4.84·26-s − 27-s − 0.968·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.66·5-s − 0.408·6-s − 0.366·7-s + 0.353·8-s + 0.333·9-s + 1.17·10-s + 1.06·11-s − 0.288·12-s − 1.34·13-s − 0.258·14-s − 0.961·15-s + 0.250·16-s + 1.96·17-s + 0.235·18-s − 1.15·19-s + 0.833·20-s + 0.211·21-s + 0.752·22-s − 0.893·23-s − 0.204·24-s + 1.77·25-s − 0.950·26-s − 0.192·27-s − 0.183·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.446234794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.446234794\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 - 3.72T + 5T^{2} \) |
| 7 | \( 1 + 0.968T + 7T^{2} \) |
| 11 | \( 1 - 3.53T + 11T^{2} \) |
| 13 | \( 1 + 4.84T + 13T^{2} \) |
| 17 | \( 1 - 8.09T + 17T^{2} \) |
| 19 | \( 1 + 5.04T + 19T^{2} \) |
| 23 | \( 1 + 4.28T + 23T^{2} \) |
| 29 | \( 1 + 5.68T + 29T^{2} \) |
| 31 | \( 1 - 7.47T + 31T^{2} \) |
| 37 | \( 1 - 6.49T + 37T^{2} \) |
| 41 | \( 1 - 1.03T + 41T^{2} \) |
| 43 | \( 1 + 5.87T + 43T^{2} \) |
| 47 | \( 1 - 6.09T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 9.72T + 59T^{2} \) |
| 61 | \( 1 + 7.33T + 61T^{2} \) |
| 67 | \( 1 + 1.88T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 4.90T + 73T^{2} \) |
| 79 | \( 1 + 1.67T + 79T^{2} \) |
| 83 | \( 1 - 2.69T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 + 3.25T + 97T^{2} \) |
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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
−10.40125776857266960007038566187, −9.951345503165018995625713647641, −9.261393329985712853462522075845, −7.72302655347767009820165839132, −6.59735369065039826239752339314, −6.01766905917368950046142644674, −5.31385985570972310647654872901, −4.20255221219213973554952413751, −2.73514186374206323488220583791, −1.54618802331854911508425578119,
1.54618802331854911508425578119, 2.73514186374206323488220583791, 4.20255221219213973554952413751, 5.31385985570972310647654872901, 6.01766905917368950046142644674, 6.59735369065039826239752339314, 7.72302655347767009820165839132, 9.261393329985712853462522075845, 9.951345503165018995625713647641, 10.40125776857266960007038566187
