L(s) = 1 | − 2.04·2-s − 3-s + 2.16·4-s − 3.68·5-s + 2.04·6-s − 0.345·8-s + 9-s + 7.53·10-s + 0.232·11-s − 2.16·12-s + 2.59·13-s + 3.68·15-s − 3.63·16-s + 4.71·17-s − 2.04·18-s − 7.74·19-s − 8.00·20-s − 0.474·22-s + 8.48·23-s + 0.345·24-s + 8.61·25-s − 5.29·26-s − 27-s − 4.10·29-s − 7.53·30-s − 10.4·31-s + 8.10·32-s + ⋯ |
L(s) = 1 | − 1.44·2-s − 0.577·3-s + 1.08·4-s − 1.64·5-s + 0.833·6-s − 0.122·8-s + 0.333·9-s + 2.38·10-s + 0.0700·11-s − 0.626·12-s + 0.718·13-s + 0.952·15-s − 0.908·16-s + 1.14·17-s − 0.481·18-s − 1.77·19-s − 1.78·20-s − 0.101·22-s + 1.76·23-s + 0.0704·24-s + 1.72·25-s − 1.03·26-s − 0.192·27-s − 0.763·29-s − 1.37·30-s − 1.87·31-s + 1.43·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 2 | \( 1 + 2.04T + 2T^{2} \) |
| 5 | \( 1 + 3.68T + 5T^{2} \) |
| 11 | \( 1 - 0.232T + 11T^{2} \) |
| 13 | \( 1 - 2.59T + 13T^{2} \) |
| 17 | \( 1 - 4.71T + 17T^{2} \) |
| 19 | \( 1 + 7.74T + 19T^{2} \) |
| 23 | \( 1 - 8.48T + 23T^{2} \) |
| 29 | \( 1 + 4.10T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + 8.13T + 37T^{2} \) |
| 43 | \( 1 - 2.59T + 43T^{2} \) |
| 47 | \( 1 - 3.41T + 47T^{2} \) |
| 53 | \( 1 + 3.35T + 53T^{2} \) |
| 59 | \( 1 + 8.02T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 - 6.60T + 67T^{2} \) |
| 71 | \( 1 - 3.23T + 71T^{2} \) |
| 73 | \( 1 + 6.63T + 73T^{2} \) |
| 79 | \( 1 + 5.24T + 79T^{2} \) |
| 83 | \( 1 - 2.50T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 + 7.90T + 97T^{2} \) |
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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
−7.77802764511301759225809801349, −7.18837691256022429585705343307, −6.78570860062849388276786921476, −5.68340019973410636425106635633, −4.78326932960418792596902161754, −3.95045964978330973695795174070, −3.31713735747414490728869201772, −1.86917610823020553422501670770, −0.862151950127140293234557283900, 0,
0.862151950127140293234557283900, 1.86917610823020553422501670770, 3.31713735747414490728869201772, 3.95045964978330973695795174070, 4.78326932960418792596902161754, 5.68340019973410636425106635633, 6.78570860062849388276786921476, 7.18837691256022429585705343307, 7.77802764511301759225809801349