| L(s) = 1 | + 0.264·2-s − 2.96·3-s − 1.93·4-s − 0.784·6-s + 2.14·7-s − 1.03·8-s + 5.81·9-s + 1.13·11-s + 5.72·12-s − 0.737·13-s + 0.566·14-s + 3.58·16-s − 2.91·17-s + 1.53·18-s + 6.85·19-s − 6.37·21-s + 0.299·22-s − 7.06·23-s + 3.08·24-s − 0.194·26-s − 8.34·27-s − 4.14·28-s − 7.65·29-s − 31-s + 3.02·32-s − 3.36·33-s − 0.769·34-s + ⋯ |
| L(s) = 1 | + 0.186·2-s − 1.71·3-s − 0.965·4-s − 0.320·6-s + 0.811·7-s − 0.367·8-s + 1.93·9-s + 0.341·11-s + 1.65·12-s − 0.204·13-s + 0.151·14-s + 0.896·16-s − 0.706·17-s + 0.361·18-s + 1.57·19-s − 1.39·21-s + 0.0638·22-s − 1.47·23-s + 0.629·24-s − 0.0382·26-s − 1.60·27-s − 0.782·28-s − 1.42·29-s − 0.179·31-s + 0.534·32-s − 0.585·33-s − 0.131·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 0.264T + 2T^{2} \) |
| 3 | \( 1 + 2.96T + 3T^{2} \) |
| 7 | \( 1 - 2.14T + 7T^{2} \) |
| 11 | \( 1 - 1.13T + 11T^{2} \) |
| 13 | \( 1 + 0.737T + 13T^{2} \) |
| 17 | \( 1 + 2.91T + 17T^{2} \) |
| 19 | \( 1 - 6.85T + 19T^{2} \) |
| 23 | \( 1 + 7.06T + 23T^{2} \) |
| 29 | \( 1 + 7.65T + 29T^{2} \) |
| 37 | \( 1 + 5.57T + 37T^{2} \) |
| 41 | \( 1 - 3.02T + 41T^{2} \) |
| 43 | \( 1 + 3.79T + 43T^{2} \) |
| 47 | \( 1 + 7.55T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 + 9.35T + 59T^{2} \) |
| 61 | \( 1 + 8.38T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 5.10T + 73T^{2} \) |
| 79 | \( 1 - 5.76T + 79T^{2} \) |
| 83 | \( 1 - 1.01T + 83T^{2} \) |
| 89 | \( 1 + 6.78T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
| 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
−9.976780106396289002156976386760, −9.260923559786790173136428591752, −8.070467773950456521520189729220, −7.16116278133791630604319930639, −6.02288256333900140107880761569, −5.34276160177294236312215964635, −4.69117326292516889909829519424, −3.77061056856297423748257253701, −1.50079479847735712555262717094, 0,
1.50079479847735712555262717094, 3.77061056856297423748257253701, 4.69117326292516889909829519424, 5.34276160177294236312215964635, 6.02288256333900140107880761569, 7.16116278133791630604319930639, 8.070467773950456521520189729220, 9.260923559786790173136428591752, 9.976780106396289002156976386760
