| L(s) = 1 | − 2.81·2-s + 3-s + 5.90·4-s − 1.40·5-s − 2.81·6-s + 4.16·7-s − 10.9·8-s + 9-s + 3.94·10-s + 5.45·11-s + 5.90·12-s + 13-s − 11.7·14-s − 1.40·15-s + 19.0·16-s − 6.38·17-s − 2.81·18-s − 0.521·19-s − 8.28·20-s + 4.16·21-s − 15.3·22-s + 9.40·23-s − 10.9·24-s − 3.02·25-s − 2.81·26-s + 27-s + 24.6·28-s + ⋯ |
| L(s) = 1 | − 1.98·2-s + 0.577·3-s + 2.95·4-s − 0.628·5-s − 1.14·6-s + 1.57·7-s − 3.87·8-s + 0.333·9-s + 1.24·10-s + 1.64·11-s + 1.70·12-s + 0.277·13-s − 3.13·14-s − 0.362·15-s + 4.75·16-s − 1.54·17-s − 0.662·18-s − 0.119·19-s − 1.85·20-s + 0.909·21-s − 3.27·22-s + 1.96·23-s − 2.23·24-s − 0.605·25-s − 0.551·26-s + 0.192·27-s + 4.64·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.206810523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.206810523\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 2.81T + 2T^{2} \) |
| 5 | \( 1 + 1.40T + 5T^{2} \) |
| 7 | \( 1 - 4.16T + 7T^{2} \) |
| 11 | \( 1 - 5.45T + 11T^{2} \) |
| 17 | \( 1 + 6.38T + 17T^{2} \) |
| 19 | \( 1 + 0.521T + 19T^{2} \) |
| 23 | \( 1 - 9.40T + 23T^{2} \) |
| 29 | \( 1 + 7.26T + 29T^{2} \) |
| 31 | \( 1 + 1.02T + 31T^{2} \) |
| 37 | \( 1 - 3.74T + 37T^{2} \) |
| 41 | \( 1 - 4.43T + 41T^{2} \) |
| 43 | \( 1 - 6.36T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 0.593T + 53T^{2} \) |
| 59 | \( 1 + 0.780T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 + 4.99T + 67T^{2} \) |
| 71 | \( 1 - 3.87T + 71T^{2} \) |
| 73 | \( 1 - 5.56T + 73T^{2} \) |
| 79 | \( 1 - 4.33T + 79T^{2} \) |
| 83 | \( 1 + 9.18T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−8.659106687215085270454601103686, −7.78438585884465154981895536188, −7.40913758069114892062845902439, −6.73577812152840814998728526940, −5.86259696379785305358246971936, −4.48289876011019010977454281310, −3.63765006640692241222797993048, −2.41502949191822520582698759048, −1.67489240616714894911076683628, −0.881588713362772619121395774571,
0.881588713362772619121395774571, 1.67489240616714894911076683628, 2.41502949191822520582698759048, 3.63765006640692241222797993048, 4.48289876011019010977454281310, 5.86259696379785305358246971936, 6.73577812152840814998728526940, 7.40913758069114892062845902439, 7.78438585884465154981895536188, 8.659106687215085270454601103686
