L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 2·7-s + 8-s + 9-s + 11-s + 12-s + 4·13-s + 2·14-s + 16-s − 17-s + 18-s − 8·19-s + 2·21-s + 22-s + 6·23-s + 24-s − 5·25-s + 4·26-s + 27-s + 2·28-s − 2·29-s + 4·31-s + 32-s + 33-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s + 1.10·13-s + 0.534·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 1.83·19-s + 0.436·21-s + 0.213·22-s + 1.25·23-s + 0.204·24-s − 25-s + 0.784·26-s + 0.192·27-s + 0.377·28-s − 0.371·29-s + 0.718·31-s + 0.176·32-s + 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.429147609\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.429147609\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−9.891765499192151571070821451946, −8.672521011063548817745639011184, −8.394919814891872206344405039084, −7.24307787406498561141886612766, −6.45079193192667644204307137635, −5.51724220764122997344606384706, −4.39565343248400556562616797524, −3.81405429750779206370242319895, −2.54966282175540929443461631671, −1.49883692542636564585418200145,
1.49883692542636564585418200145, 2.54966282175540929443461631671, 3.81405429750779206370242319895, 4.39565343248400556562616797524, 5.51724220764122997344606384706, 6.45079193192667644204307137635, 7.24307787406498561141886612766, 8.394919814891872206344405039084, 8.672521011063548817745639011184, 9.891765499192151571070821451946