L(s) = 1 | + 0.741·2-s − 1.44·4-s − 5-s − 3.30·7-s − 2.55·8-s − 0.741·10-s + 1.81·13-s − 2.44·14-s + 1.00·16-s + 3.30·17-s − 1.48·19-s + 1.44·20-s + 4.89·23-s + 25-s + 1.34·26-s + 4.78·28-s + 8.08·29-s − 2·31-s + 5.86·32-s + 2.44·34-s + 3.30·35-s + 2.89·37-s − 1.10·38-s + 2.55·40-s − 5.11·41-s + 3.30·43-s + 3.63·46-s + ⋯ |
L(s) = 1 | + 0.524·2-s − 0.724·4-s − 0.447·5-s − 1.24·7-s − 0.904·8-s − 0.234·10-s + 0.504·13-s − 0.654·14-s + 0.250·16-s + 0.800·17-s − 0.340·19-s + 0.324·20-s + 1.02·23-s + 0.200·25-s + 0.264·26-s + 0.904·28-s + 1.50·29-s − 0.359·31-s + 1.03·32-s + 0.420·34-s + 0.558·35-s + 0.476·37-s − 0.178·38-s + 0.404·40-s − 0.799·41-s + 0.503·43-s + 0.535·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 0.741T + 2T^{2} \) |
| 7 | \( 1 + 3.30T + 7T^{2} \) |
| 13 | \( 1 - 1.81T + 13T^{2} \) |
| 17 | \( 1 - 3.30T + 17T^{2} \) |
| 19 | \( 1 + 1.48T + 19T^{2} \) |
| 23 | \( 1 - 4.89T + 23T^{2} \) |
| 29 | \( 1 - 8.08T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 2.89T + 37T^{2} \) |
| 41 | \( 1 + 5.11T + 41T^{2} \) |
| 43 | \( 1 - 3.30T + 43T^{2} \) |
| 47 | \( 1 + 9.79T + 47T^{2} \) |
| 53 | \( 1 + 1.10T + 53T^{2} \) |
| 59 | \( 1 + 1.10T + 59T^{2} \) |
| 61 | \( 1 - 9.57T + 61T^{2} \) |
| 67 | \( 1 + 0.898T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 17.6T + 79T^{2} \) |
| 83 | \( 1 + 4.78T + 83T^{2} \) |
| 89 | \( 1 - 15.7T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−7.86725503529945546445740315309, −6.91250396033923775360842178126, −6.32120269292809431133693154448, −5.61705980175067973614507497083, −4.80224827357890861900771394492, −4.07790753065333851054698253189, −3.27396086203878247747772280890, −2.88606703393221000419150491996, −1.12547456000065746731045234235, 0,
1.12547456000065746731045234235, 2.88606703393221000419150491996, 3.27396086203878247747772280890, 4.07790753065333851054698253189, 4.80224827357890861900771394492, 5.61705980175067973614507497083, 6.32120269292809431133693154448, 6.91250396033923775360842178126, 7.86725503529945546445740315309