| L(s) = 1 | + 5.09·2-s − 5·3-s + 17.9·4-s + 5·5-s − 25.4·6-s + 20.3·7-s + 50.9·8-s − 2·9-s + 25.4·10-s − 89.9·12-s + 61.1·13-s + 103.·14-s − 25·15-s + 115.·16-s − 20.3·17-s − 10.1·18-s − 101.·19-s + 89.9·20-s − 101.·21-s + 35·23-s − 254.·24-s − 100·25-s + 312·26-s + 145·27-s + 367.·28-s − 203.·29-s − 127.·30-s + ⋯ |
| L(s) = 1 | + 1.80·2-s − 0.962·3-s + 2.24·4-s + 0.447·5-s − 1.73·6-s + 1.10·7-s + 2.25·8-s − 0.0740·9-s + 0.806·10-s − 2.16·12-s + 1.30·13-s + 1.98·14-s − 0.430·15-s + 1.81·16-s − 0.290·17-s − 0.133·18-s − 1.23·19-s + 1.00·20-s − 1.05·21-s + 0.317·23-s − 2.16·24-s − 0.800·25-s + 2.35·26-s + 1.03·27-s + 2.47·28-s − 1.30·29-s − 0.775·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.786351985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.786351985\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 - 5.09T + 8T^{2} \) |
| 3 | \( 1 + 5T + 27T^{2} \) |
| 5 | \( 1 - 5T + 125T^{2} \) |
| 7 | \( 1 - 20.3T + 343T^{2} \) |
| 13 | \( 1 - 61.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 20.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 101.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 35T + 1.21e4T^{2} \) |
| 29 | \( 1 + 203.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 15T + 2.97e4T^{2} \) |
| 37 | \( 1 + 265T + 5.06e4T^{2} \) |
| 41 | \( 1 + 101.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 448.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 380T + 1.03e5T^{2} \) |
| 53 | \( 1 - 510T + 1.48e5T^{2} \) |
| 59 | \( 1 - 21T + 2.05e5T^{2} \) |
| 61 | \( 1 - 203.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 585T + 3.00e5T^{2} \) |
| 71 | \( 1 - 313T + 3.57e5T^{2} \) |
| 73 | \( 1 - 469.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 611.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 652.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 185T + 7.04e5T^{2} \) |
| 97 | \( 1 - 785T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.11288451546448122703602777558, −11.97861359686842933348632449821, −11.23471617863429873179670355307, −10.65900543321220122293120090798, −8.471216563040890018729496839514, −6.78278335259548599839509815879, −5.83777879973950033031614491160, −5.10287415509003810677952181145, −3.86640822682024360043106172384, −1.92073020191615911972540918959,
1.92073020191615911972540918959, 3.86640822682024360043106172384, 5.10287415509003810677952181145, 5.83777879973950033031614491160, 6.78278335259548599839509815879, 8.471216563040890018729496839514, 10.65900543321220122293120090798, 11.23471617863429873179670355307, 11.97861359686842933348632449821, 13.11288451546448122703602777558
