| L(s) = 1 | + 0.719·2-s + 0.234·3-s − 1.48·4-s + 0.168·6-s − 7-s − 2.50·8-s − 2.94·9-s + 11-s − 0.347·12-s − 2.88·13-s − 0.719·14-s + 1.16·16-s + 5.70·17-s − 2.11·18-s − 0.482·19-s − 0.234·21-s + 0.719·22-s + 2.02·23-s − 0.587·24-s − 2.07·26-s − 1.39·27-s + 1.48·28-s + 6.27·29-s + 4.19·31-s + 5.84·32-s + 0.234·33-s + 4.10·34-s + ⋯ |
| L(s) = 1 | + 0.508·2-s + 0.135·3-s − 0.741·4-s + 0.0688·6-s − 0.377·7-s − 0.885·8-s − 0.981·9-s + 0.301·11-s − 0.100·12-s − 0.800·13-s − 0.192·14-s + 0.290·16-s + 1.38·17-s − 0.499·18-s − 0.110·19-s − 0.0511·21-s + 0.153·22-s + 0.421·23-s − 0.119·24-s − 0.407·26-s − 0.268·27-s + 0.280·28-s + 1.16·29-s + 0.752·31-s + 1.03·32-s + 0.0408·33-s + 0.703·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.513694164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.513694164\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 0.719T + 2T^{2} \) |
| 3 | \( 1 - 0.234T + 3T^{2} \) |
| 13 | \( 1 + 2.88T + 13T^{2} \) |
| 17 | \( 1 - 5.70T + 17T^{2} \) |
| 19 | \( 1 + 0.482T + 19T^{2} \) |
| 23 | \( 1 - 2.02T + 23T^{2} \) |
| 29 | \( 1 - 6.27T + 29T^{2} \) |
| 31 | \( 1 - 4.19T + 31T^{2} \) |
| 37 | \( 1 + 0.237T + 37T^{2} \) |
| 41 | \( 1 + 0.0283T + 41T^{2} \) |
| 43 | \( 1 - 2.97T + 43T^{2} \) |
| 47 | \( 1 - 6.61T + 47T^{2} \) |
| 53 | \( 1 - 3.48T + 53T^{2} \) |
| 59 | \( 1 + 1.79T + 59T^{2} \) |
| 61 | \( 1 - 4.29T + 61T^{2} \) |
| 67 | \( 1 + 5.58T + 67T^{2} \) |
| 71 | \( 1 - 15.0T + 71T^{2} \) |
| 73 | \( 1 + 4.18T + 73T^{2} \) |
| 79 | \( 1 + 4.44T + 79T^{2} \) |
| 83 | \( 1 + 2.19T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 7.13T + 97T^{2} \) |
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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
−9.201660246683081556683480449940, −8.471835968856150773847337513897, −7.76456012458061895633045206022, −6.67994685234424065446705370211, −5.79900705301349366500411265097, −5.19613227295336068656723822202, −4.28072068992080021925292825540, −3.30714312660709484951090057143, −2.64145528928680705529160765934, −0.75654141829225319214570562645,
0.75654141829225319214570562645, 2.64145528928680705529160765934, 3.30714312660709484951090057143, 4.28072068992080021925292825540, 5.19613227295336068656723822202, 5.79900705301349366500411265097, 6.67994685234424065446705370211, 7.76456012458061895633045206022, 8.471835968856150773847337513897, 9.201660246683081556683480449940
