| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 1.23·7-s + 8-s + 9-s − 4.47·11-s − 12-s − 13-s + 1.23·14-s + 16-s − 2.76·17-s + 18-s − 7.23·19-s − 1.23·21-s − 4.47·22-s − 7.23·23-s − 24-s − 26-s − 27-s + 1.23·28-s + 9.70·29-s − 4·31-s + 32-s + 4.47·33-s − 2.76·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s + 0.467·7-s + 0.353·8-s + 0.333·9-s − 1.34·11-s − 0.288·12-s − 0.277·13-s + 0.330·14-s + 0.250·16-s − 0.670·17-s + 0.235·18-s − 1.66·19-s − 0.269·21-s − 0.953·22-s − 1.50·23-s − 0.204·24-s − 0.196·26-s − 0.192·27-s + 0.233·28-s + 1.80·29-s − 0.718·31-s + 0.176·32-s + 0.778·33-s − 0.474·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 17 | \( 1 + 2.76T + 17T^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 23 | \( 1 + 7.23T + 23T^{2} \) |
| 29 | \( 1 - 9.70T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 6.94T + 37T^{2} \) |
| 41 | \( 1 - 12.4T + 41T^{2} \) |
| 43 | \( 1 + 6.47T + 43T^{2} \) |
| 47 | \( 1 - 4.94T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 - 0.472T + 59T^{2} \) |
| 61 | \( 1 - 6.94T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 6.47T + 71T^{2} \) |
| 73 | \( 1 + 8.76T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 + 8.47T + 89T^{2} \) |
| 97 | \( 1 + 9.70T + 97T^{2} \) |
| 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
−8.530779694568573213995301552174, −8.004332164035201635894589716494, −7.03845459460414655963082095928, −6.26948536369219982766681897972, −5.51991667166007308496010389151, −4.69024563761773266461927590004, −4.12832467108517016568004614766, −2.71788715017534627395376652419, −1.88660992270747490487052641091, 0,
1.88660992270747490487052641091, 2.71788715017534627395376652419, 4.12832467108517016568004614766, 4.69024563761773266461927590004, 5.51991667166007308496010389151, 6.26948536369219982766681897972, 7.03845459460414655963082095928, 8.004332164035201635894589716494, 8.530779694568573213995301552174
