L(s) = 1 | + 1.22·2-s + 3-s − 0.492·4-s + 2.78·5-s + 1.22·6-s − 7-s − 3.06·8-s + 9-s + 3.41·10-s + 0.773·11-s − 0.492·12-s + 4.48·13-s − 1.22·14-s + 2.78·15-s − 2.77·16-s − 0.115·17-s + 1.22·18-s + 3.33·19-s − 1.37·20-s − 21-s + 0.949·22-s − 1.55·23-s − 3.06·24-s + 2.75·25-s + 5.50·26-s + 27-s + 0.492·28-s + ⋯ |
L(s) = 1 | + 0.868·2-s + 0.577·3-s − 0.246·4-s + 1.24·5-s + 0.501·6-s − 0.377·7-s − 1.08·8-s + 0.333·9-s + 1.08·10-s + 0.233·11-s − 0.142·12-s + 1.24·13-s − 0.328·14-s + 0.718·15-s − 0.693·16-s − 0.0281·17-s + 0.289·18-s + 0.765·19-s − 0.306·20-s − 0.218·21-s + 0.202·22-s − 0.323·23-s − 0.624·24-s + 0.550·25-s + 1.08·26-s + 0.192·27-s + 0.0930·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.203644056\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.203644056\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 + T \) |
good | 2 | \( 1 - 1.22T + 2T^{2} \) |
| 5 | \( 1 - 2.78T + 5T^{2} \) |
| 11 | \( 1 - 0.773T + 11T^{2} \) |
| 13 | \( 1 - 4.48T + 13T^{2} \) |
| 17 | \( 1 + 0.115T + 17T^{2} \) |
| 19 | \( 1 - 3.33T + 19T^{2} \) |
| 23 | \( 1 + 1.55T + 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 + 3.61T + 31T^{2} \) |
| 37 | \( 1 - 4.37T + 37T^{2} \) |
| 41 | \( 1 - 9.83T + 41T^{2} \) |
| 43 | \( 1 + 5.70T + 43T^{2} \) |
| 47 | \( 1 + 6.40T + 47T^{2} \) |
| 53 | \( 1 + 6.08T + 53T^{2} \) |
| 59 | \( 1 + 2.05T + 59T^{2} \) |
| 61 | \( 1 - 6.27T + 61T^{2} \) |
| 67 | \( 1 + 7.09T + 67T^{2} \) |
| 71 | \( 1 - 5.03T + 71T^{2} \) |
| 73 | \( 1 - 16.7T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 - 17.8T + 83T^{2} \) |
| 89 | \( 1 - 3.27T + 89T^{2} \) |
| 97 | \( 1 - 8.04T + 97T^{2} \) |
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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
−8.571558270545469950201350182246, −7.80249501313535217871645475679, −6.53388085232892627630672188413, −6.21937251402875273598019092899, −5.46746148964457509163376762374, −4.67990819254670371518014209831, −3.75271928830335179034527404400, −3.14201873222716169497921815024, −2.22183277501431521940011196995, −1.07185294059287906269984293607,
1.07185294059287906269984293607, 2.22183277501431521940011196995, 3.14201873222716169497921815024, 3.75271928830335179034527404400, 4.67990819254670371518014209831, 5.46746148964457509163376762374, 6.21937251402875273598019092899, 6.53388085232892627630672188413, 7.80249501313535217871645475679, 8.571558270545469950201350182246