L(s) = 1 | − 2-s + 0.815·3-s + 4-s − 5-s − 0.815·6-s + 0.327·7-s − 8-s − 2.33·9-s + 10-s + 2.28·11-s + 0.815·12-s + 13-s − 0.327·14-s − 0.815·15-s + 16-s + 1.51·17-s + 2.33·18-s + 2.59·19-s − 20-s + 0.267·21-s − 2.28·22-s − 5.27·23-s − 0.815·24-s + 25-s − 26-s − 4.35·27-s + 0.327·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.470·3-s + 0.5·4-s − 0.447·5-s − 0.333·6-s + 0.123·7-s − 0.353·8-s − 0.778·9-s + 0.316·10-s + 0.688·11-s + 0.235·12-s + 0.277·13-s − 0.0875·14-s − 0.210·15-s + 0.250·16-s + 0.366·17-s + 0.550·18-s + 0.595·19-s − 0.223·20-s + 0.0583·21-s − 0.486·22-s − 1.10·23-s − 0.166·24-s + 0.200·25-s − 0.196·26-s − 0.837·27-s + 0.0619·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.378381903\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.378381903\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
good | 3 | \( 1 - 0.815T + 3T^{2} \) |
| 7 | \( 1 - 0.327T + 7T^{2} \) |
| 11 | \( 1 - 2.28T + 11T^{2} \) |
| 17 | \( 1 - 1.51T + 17T^{2} \) |
| 19 | \( 1 - 2.59T + 19T^{2} \) |
| 23 | \( 1 + 5.27T + 23T^{2} \) |
| 29 | \( 1 - 4.04T + 29T^{2} \) |
| 37 | \( 1 - 0.482T + 37T^{2} \) |
| 41 | \( 1 - 3.28T + 41T^{2} \) |
| 43 | \( 1 - 6.93T + 43T^{2} \) |
| 47 | \( 1 - 4.58T + 47T^{2} \) |
| 53 | \( 1 - 0.858T + 53T^{2} \) |
| 59 | \( 1 - 8.54T + 59T^{2} \) |
| 61 | \( 1 + 4.89T + 61T^{2} \) |
| 67 | \( 1 + 8.76T + 67T^{2} \) |
| 71 | \( 1 - 1.67T + 71T^{2} \) |
| 73 | \( 1 + 9.74T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 - 6.40T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 + 3.12T + 97T^{2} \) |
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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
−8.550771068524618359230095915314, −7.77449421207650009976179446488, −7.31366717378776597770793604481, −6.23514497749956028519866572764, −5.73818146297794546616844878199, −4.54379151313787628650078117250, −3.63779019417885253765017807709, −2.92126540339807743601241141129, −1.90406972014929334232721213941, −0.73075150433910962293179267423,
0.73075150433910962293179267423, 1.90406972014929334232721213941, 2.92126540339807743601241141129, 3.63779019417885253765017807709, 4.54379151313787628650078117250, 5.73818146297794546616844878199, 6.23514497749956028519866572764, 7.31366717378776597770793604481, 7.77449421207650009976179446488, 8.550771068524618359230095915314