| L(s) = 1 | − 1.73·2-s + 3-s + 1.00·4-s + 2.89·5-s − 1.73·6-s + 3.30·7-s + 1.72·8-s + 9-s − 5.01·10-s + 3.25·11-s + 1.00·12-s − 1.42·13-s − 5.72·14-s + 2.89·15-s − 4.99·16-s + 17-s − 1.73·18-s + 2.32·19-s + 2.90·20-s + 3.30·21-s − 5.63·22-s − 8.40·23-s + 1.72·24-s + 3.38·25-s + 2.46·26-s + 27-s + 3.30·28-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.577·3-s + 0.501·4-s + 1.29·5-s − 0.707·6-s + 1.24·7-s + 0.611·8-s + 0.333·9-s − 1.58·10-s + 0.980·11-s + 0.289·12-s − 0.395·13-s − 1.52·14-s + 0.747·15-s − 1.24·16-s + 0.242·17-s − 0.408·18-s + 0.532·19-s + 0.648·20-s + 0.720·21-s − 1.20·22-s − 1.75·23-s + 0.352·24-s + 0.676·25-s + 0.484·26-s + 0.192·27-s + 0.625·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.012927359\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.012927359\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 5 | \( 1 - 2.89T + 5T^{2} \) |
| 7 | \( 1 - 3.30T + 7T^{2} \) |
| 11 | \( 1 - 3.25T + 11T^{2} \) |
| 13 | \( 1 + 1.42T + 13T^{2} \) |
| 19 | \( 1 - 2.32T + 19T^{2} \) |
| 23 | \( 1 + 8.40T + 23T^{2} \) |
| 29 | \( 1 + 0.650T + 29T^{2} \) |
| 31 | \( 1 + 0.747T + 31T^{2} \) |
| 37 | \( 1 - 5.76T + 37T^{2} \) |
| 41 | \( 1 + 9.70T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 - 6.77T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 + 7.73T + 59T^{2} \) |
| 61 | \( 1 + 2.72T + 61T^{2} \) |
| 67 | \( 1 + 2.41T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 5.37T + 73T^{2} \) |
| 83 | \( 1 + 4.72T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 + 2.34T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.656710441156363992055708625426, −7.79267719132053541285515749837, −7.40680439032911315754710937589, −6.36195290216377980752568149930, −5.57470738482218669565557177896, −4.66094238934701030666799970529, −3.87545238554676857299766430761, −2.35031798799869208435317772717, −1.82448267745375259551119538119, −1.04740337062937138399439037337,
1.04740337062937138399439037337, 1.82448267745375259551119538119, 2.35031798799869208435317772717, 3.87545238554676857299766430761, 4.66094238934701030666799970529, 5.57470738482218669565557177896, 6.36195290216377980752568149930, 7.40680439032911315754710937589, 7.79267719132053541285515749837, 8.656710441156363992055708625426
