L(s) = 1 | − 2-s − 0.323·3-s + 4-s + 5-s + 0.323·6-s + 4.27·7-s − 8-s − 2.89·9-s − 10-s − 6.20·11-s − 0.323·12-s + 1.40·13-s − 4.27·14-s − 0.323·15-s + 16-s + 4.33·17-s + 2.89·18-s − 2.64·19-s + 20-s − 1.38·21-s + 6.20·22-s − 4.01·23-s + 0.323·24-s + 25-s − 1.40·26-s + 1.90·27-s + 4.27·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.186·3-s + 0.5·4-s + 0.447·5-s + 0.132·6-s + 1.61·7-s − 0.353·8-s − 0.965·9-s − 0.316·10-s − 1.87·11-s − 0.0933·12-s + 0.389·13-s − 1.14·14-s − 0.0834·15-s + 0.250·16-s + 1.05·17-s + 0.682·18-s − 0.607·19-s + 0.223·20-s − 0.301·21-s + 1.32·22-s − 0.836·23-s + 0.0660·24-s + 0.200·25-s − 0.275·26-s + 0.366·27-s + 0.807·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.299307211\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.299307211\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
good | 3 | \( 1 + 0.323T + 3T^{2} \) |
| 7 | \( 1 - 4.27T + 7T^{2} \) |
| 11 | \( 1 + 6.20T + 11T^{2} \) |
| 13 | \( 1 - 1.40T + 13T^{2} \) |
| 17 | \( 1 - 4.33T + 17T^{2} \) |
| 19 | \( 1 + 2.64T + 19T^{2} \) |
| 23 | \( 1 + 4.01T + 23T^{2} \) |
| 29 | \( 1 + 5.35T + 29T^{2} \) |
| 31 | \( 1 + 3.65T + 31T^{2} \) |
| 37 | \( 1 - 4.46T + 37T^{2} \) |
| 41 | \( 1 + 5.68T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 - 2.41T + 53T^{2} \) |
| 59 | \( 1 - 2.54T + 59T^{2} \) |
| 61 | \( 1 - 4.37T + 61T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 - 6.97T + 79T^{2} \) |
| 83 | \( 1 - 6.32T + 83T^{2} \) |
| 89 | \( 1 + 0.107T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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
−8.324229673896394687640594805396, −7.86835338427270711865383192057, −7.36436609843317880136906850346, −6.01710533381916770387800678877, −5.52701569069060310641783545143, −5.02481324046171399241782587216, −3.78314490913184863020299429703, −2.51296385651428750090168371923, −2.03188013708402973729064789329, −0.71969028736183347835621649904,
0.71969028736183347835621649904, 2.03188013708402973729064789329, 2.51296385651428750090168371923, 3.78314490913184863020299429703, 5.02481324046171399241782587216, 5.52701569069060310641783545143, 6.01710533381916770387800678877, 7.36436609843317880136906850346, 7.86835338427270711865383192057, 8.324229673896394687640594805396