L(s) = 1 | − 3-s − 1.73·5-s − 0.732·7-s + 9-s − 0.732·11-s + 1.73·15-s − 3.73·17-s − 1.26·19-s + 0.732·21-s − 2.19·23-s − 2.00·25-s − 27-s − 5.92·29-s + 5.46·31-s + 0.732·33-s + 1.26·35-s + 8.46·37-s + 5·41-s + 1.80·43-s − 1.73·45-s − 6.73·47-s − 6.46·49-s + 3.73·51-s + 5.92·53-s + 1.26·55-s + 1.26·57-s − 9.73·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.774·5-s − 0.276·7-s + 0.333·9-s − 0.220·11-s + 0.447·15-s − 0.905·17-s − 0.290·19-s + 0.159·21-s − 0.457·23-s − 0.400·25-s − 0.192·27-s − 1.10·29-s + 0.981·31-s + 0.127·33-s + 0.214·35-s + 1.39·37-s + 0.780·41-s + 0.275·43-s − 0.258·45-s − 0.981·47-s − 0.923·49-s + 0.522·51-s + 0.814·53-s + 0.170·55-s + 0.167·57-s − 1.24·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7659375060\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7659375060\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 1.73T + 5T^{2} \) |
| 7 | \( 1 + 0.732T + 7T^{2} \) |
| 11 | \( 1 + 0.732T + 11T^{2} \) |
| 17 | \( 1 + 3.73T + 17T^{2} \) |
| 19 | \( 1 + 1.26T + 19T^{2} \) |
| 23 | \( 1 + 2.19T + 23T^{2} \) |
| 29 | \( 1 + 5.92T + 29T^{2} \) |
| 31 | \( 1 - 5.46T + 31T^{2} \) |
| 37 | \( 1 - 8.46T + 37T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 - 1.80T + 43T^{2} \) |
| 47 | \( 1 + 6.73T + 47T^{2} \) |
| 53 | \( 1 - 5.92T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 9.73T + 61T^{2} \) |
| 67 | \( 1 - 5.26T + 67T^{2} \) |
| 71 | \( 1 + 8.19T + 71T^{2} \) |
| 73 | \( 1 + 1.19T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 + 2.53T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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
−8.191372681695195403510959301567, −7.80803906115713024551545492286, −6.90102600209994939858648622316, −6.26413670702802987310040899650, −5.53573721430042987338957166587, −4.49891893266247674554683403926, −4.07016648712750188671592046578, −3.00726265000428923448322928202, −1.93590722692513333375223651863, −0.49844945307182597708130948999,
0.49844945307182597708130948999, 1.93590722692513333375223651863, 3.00726265000428923448322928202, 4.07016648712750188671592046578, 4.49891893266247674554683403926, 5.53573721430042987338957166587, 6.26413670702802987310040899650, 6.90102600209994939858648622316, 7.80803906115713024551545492286, 8.191372681695195403510959301567