| L(s) = 1 | − 2-s − 3.14·3-s + 4-s − 1.53·5-s + 3.14·6-s − 4.82·7-s − 8-s + 6.91·9-s + 1.53·10-s − 1.09·11-s − 3.14·12-s − 1.76·13-s + 4.82·14-s + 4.82·15-s + 16-s − 2.94·17-s − 6.91·18-s + 0.825·19-s − 1.53·20-s + 15.1·21-s + 1.09·22-s + 0.243·23-s + 3.14·24-s − 2.65·25-s + 1.76·26-s − 12.3·27-s − 4.82·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.81·3-s + 0.5·4-s − 0.685·5-s + 1.28·6-s − 1.82·7-s − 0.353·8-s + 2.30·9-s + 0.484·10-s − 0.329·11-s − 0.909·12-s − 0.490·13-s + 1.28·14-s + 1.24·15-s + 0.250·16-s − 0.714·17-s − 1.63·18-s + 0.189·19-s − 0.342·20-s + 3.31·21-s + 0.233·22-s + 0.0507·23-s + 0.642·24-s − 0.530·25-s + 0.347·26-s − 2.37·27-s − 0.911·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 + 3.14T + 3T^{2} \) |
| 5 | \( 1 + 1.53T + 5T^{2} \) |
| 7 | \( 1 + 4.82T + 7T^{2} \) |
| 11 | \( 1 + 1.09T + 11T^{2} \) |
| 13 | \( 1 + 1.76T + 13T^{2} \) |
| 17 | \( 1 + 2.94T + 17T^{2} \) |
| 19 | \( 1 - 0.825T + 19T^{2} \) |
| 23 | \( 1 - 0.243T + 23T^{2} \) |
| 29 | \( 1 - 5.57T + 29T^{2} \) |
| 31 | \( 1 - 5.73T + 31T^{2} \) |
| 41 | \( 1 - 2.91T + 41T^{2} \) |
| 43 | \( 1 + 4.37T + 43T^{2} \) |
| 47 | \( 1 - 2.26T + 47T^{2} \) |
| 53 | \( 1 - 6.14T + 53T^{2} \) |
| 59 | \( 1 + 8.82T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 8.98T + 67T^{2} \) |
| 71 | \( 1 + 1.21T + 71T^{2} \) |
| 73 | \( 1 - 8.79T + 73T^{2} \) |
| 79 | \( 1 + 2.15T + 79T^{2} \) |
| 83 | \( 1 - 5.09T + 83T^{2} \) |
| 89 | \( 1 + 8.97T + 89T^{2} \) |
| 97 | \( 1 + 4.35T + 97T^{2} \) |
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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
−8.436850414047934646596594544631, −7.41879880395851295955453490262, −6.78089294168477005083307576833, −6.35587375030501739358156246253, −5.57889190392409470898850184620, −4.61144764799134516092484520368, −3.69425275720894417602372016558, −2.52800499106271298797746608018, −0.78350740980579571108281264307, 0,
0.78350740980579571108281264307, 2.52800499106271298797746608018, 3.69425275720894417602372016558, 4.61144764799134516092484520368, 5.57889190392409470898850184620, 6.35587375030501739358156246253, 6.78089294168477005083307576833, 7.41879880395851295955453490262, 8.436850414047934646596594544631
