L(s) = 1 | − 2-s − 3-s + 4-s + 0.864·5-s + 6-s + 2.90·7-s − 8-s + 9-s − 0.864·10-s − 1.74·11-s − 12-s + 4.31·13-s − 2.90·14-s − 0.864·15-s + 16-s − 3.14·17-s − 18-s + 19-s + 0.864·20-s − 2.90·21-s + 1.74·22-s + 2.44·23-s + 24-s − 4.25·25-s − 4.31·26-s − 27-s + 2.90·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.386·5-s + 0.408·6-s + 1.09·7-s − 0.353·8-s + 0.333·9-s − 0.273·10-s − 0.525·11-s − 0.288·12-s + 1.19·13-s − 0.775·14-s − 0.223·15-s + 0.250·16-s − 0.762·17-s − 0.235·18-s + 0.229·19-s + 0.193·20-s − 0.632·21-s + 0.371·22-s + 0.510·23-s + 0.204·24-s − 0.850·25-s − 0.846·26-s − 0.192·27-s + 0.548·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 - T \) |
good | 5 | \( 1 - 0.864T + 5T^{2} \) |
| 7 | \( 1 - 2.90T + 7T^{2} \) |
| 11 | \( 1 + 1.74T + 11T^{2} \) |
| 13 | \( 1 - 4.31T + 13T^{2} \) |
| 17 | \( 1 + 3.14T + 17T^{2} \) |
| 23 | \( 1 - 2.44T + 23T^{2} \) |
| 29 | \( 1 + 4.86T + 29T^{2} \) |
| 31 | \( 1 + 7.65T + 31T^{2} \) |
| 37 | \( 1 - 9.48T + 37T^{2} \) |
| 41 | \( 1 + 6.40T + 41T^{2} \) |
| 43 | \( 1 - 4.11T + 43T^{2} \) |
| 47 | \( 1 + 9.98T + 47T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 + 2.93T + 71T^{2} \) |
| 73 | \( 1 - 4.34T + 73T^{2} \) |
| 79 | \( 1 + 4.78T + 79T^{2} \) |
| 83 | \( 1 - 8.79T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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
−7.79376783555208133596597688507, −7.15716074373006269830983286215, −6.20705434645499617189246474330, −5.75125537145143788727058026215, −4.95035024708635860271599308822, −4.15780021642330244284901898908, −3.09024056528869024515454952917, −1.88942815868145833889116243813, −1.40047933535407613697192247132, 0,
1.40047933535407613697192247132, 1.88942815868145833889116243813, 3.09024056528869024515454952917, 4.15780021642330244284901898908, 4.95035024708635860271599308822, 5.75125537145143788727058026215, 6.20705434645499617189246474330, 7.15716074373006269830983286215, 7.79376783555208133596597688507