L(s) = 1 | − 0.632·2-s + 1.91·3-s − 1.59·4-s − 1.21·6-s − 4.08·7-s + 2.27·8-s + 0.685·9-s − 4.34·11-s − 3.07·12-s + 1.55·13-s + 2.58·14-s + 1.75·16-s − 6.36·17-s − 0.433·18-s − 7.84·21-s + 2.75·22-s + 3.36·23-s + 4.37·24-s − 0.981·26-s − 4.44·27-s + 6.54·28-s − 5.21·29-s − 6.56·31-s − 5.66·32-s − 8.34·33-s + 4.02·34-s − 1.09·36-s + ⋯ |
L(s) = 1 | − 0.447·2-s + 1.10·3-s − 0.799·4-s − 0.495·6-s − 1.54·7-s + 0.805·8-s + 0.228·9-s − 1.31·11-s − 0.886·12-s + 0.430·13-s + 0.691·14-s + 0.439·16-s − 1.54·17-s − 0.102·18-s − 1.71·21-s + 0.586·22-s + 0.701·23-s + 0.892·24-s − 0.192·26-s − 0.855·27-s + 1.23·28-s − 0.967·29-s − 1.17·31-s − 1.00·32-s − 1.45·33-s + 0.690·34-s − 0.182·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4786589177\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4786589177\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + 0.632T + 2T^{2} \) |
| 3 | \( 1 - 1.91T + 3T^{2} \) |
| 7 | \( 1 + 4.08T + 7T^{2} \) |
| 11 | \( 1 + 4.34T + 11T^{2} \) |
| 13 | \( 1 - 1.55T + 13T^{2} \) |
| 17 | \( 1 + 6.36T + 17T^{2} \) |
| 23 | \( 1 - 3.36T + 23T^{2} \) |
| 29 | \( 1 + 5.21T + 29T^{2} \) |
| 31 | \( 1 + 6.56T + 31T^{2} \) |
| 37 | \( 1 - 0.180T + 37T^{2} \) |
| 41 | \( 1 + 0.0257T + 41T^{2} \) |
| 43 | \( 1 + 4.57T + 43T^{2} \) |
| 47 | \( 1 - 1.43T + 47T^{2} \) |
| 53 | \( 1 + 1.60T + 53T^{2} \) |
| 59 | \( 1 + 9.54T + 59T^{2} \) |
| 61 | \( 1 + 6.09T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 + 17.2T + 79T^{2} \) |
| 83 | \( 1 - 5.15T + 83T^{2} \) |
| 89 | \( 1 - 0.507T + 89T^{2} \) |
| 97 | \( 1 - 3.28T + 97T^{2} \) |
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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
−7.893575029059828544569226517740, −7.27227454176974629706860492253, −6.52343856169682742730771508595, −5.64995883431204798260834419520, −4.93637521869340288463675305927, −3.94754051427916899339632128045, −3.41675826859915730622083438291, −2.70841140700118639496247319693, −1.89038179641265735118164407683, −0.31909071972204186832953974839,
0.31909071972204186832953974839, 1.89038179641265735118164407683, 2.70841140700118639496247319693, 3.41675826859915730622083438291, 3.94754051427916899339632128045, 4.93637521869340288463675305927, 5.64995883431204798260834419520, 6.52343856169682742730771508595, 7.27227454176974629706860492253, 7.893575029059828544569226517740