L(s) = 1 | + 2.22·2-s − 3-s + 2.95·4-s − 3.65·5-s − 2.22·6-s + 7-s + 2.12·8-s + 9-s − 8.14·10-s − 5.04·11-s − 2.95·12-s + 3.81·13-s + 2.22·14-s + 3.65·15-s − 1.17·16-s − 0.994·17-s + 2.22·18-s + 5.71·19-s − 10.8·20-s − 21-s − 11.2·22-s − 2.03·23-s − 2.12·24-s + 8.37·25-s + 8.48·26-s − 27-s + 2.95·28-s + ⋯ |
L(s) = 1 | + 1.57·2-s − 0.577·3-s + 1.47·4-s − 1.63·5-s − 0.908·6-s + 0.377·7-s + 0.752·8-s + 0.333·9-s − 2.57·10-s − 1.52·11-s − 0.853·12-s + 1.05·13-s + 0.594·14-s + 0.944·15-s − 0.293·16-s − 0.241·17-s + 0.524·18-s + 1.31·19-s − 2.41·20-s − 0.218·21-s − 2.39·22-s − 0.423·23-s − 0.434·24-s + 1.67·25-s + 1.66·26-s − 0.192·27-s + 0.558·28-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
\(L(1)\) |
\(\approx\) |
\(2.515062120\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.515062120\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.22T + 2T^{2} \) |
| 5 | \( 1 + 3.65T + 5T^{2} \) |
| 11 | \( 1 + 5.04T + 11T^{2} \) |
| 13 | \( 1 - 3.81T + 13T^{2} \) |
| 17 | \( 1 + 0.994T + 17T^{2} \) |
| 19 | \( 1 - 5.71T + 19T^{2} \) |
| 23 | \( 1 + 2.03T + 23T^{2} \) |
| 29 | \( 1 - 0.807T + 29T^{2} \) |
| 31 | \( 1 - 1.21T + 31T^{2} \) |
| 37 | \( 1 + 1.13T + 37T^{2} \) |
| 41 | \( 1 - 0.234T + 41T^{2} \) |
| 43 | \( 1 - 6.74T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 4.70T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 - 2.43T + 67T^{2} \) |
| 71 | \( 1 + 5.80T + 71T^{2} \) |
| 73 | \( 1 + 0.0792T + 73T^{2} \) |
| 79 | \( 1 - 7.06T + 79T^{2} \) |
| 83 | \( 1 + 4.00T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−8.104953483061807478708330762148, −7.54378658277298552888846776515, −6.91898053873343207079042012278, −5.89659508378136399512481015178, −5.32760877515672001922052635416, −4.65789645644323719844389861950, −3.95577757541188660038272323252, −3.35940331216820098701255100602, −2.41507446834935598862022086120, −0.72295959140129791470079058178,
0.72295959140129791470079058178, 2.41507446834935598862022086120, 3.35940331216820098701255100602, 3.95577757541188660038272323252, 4.65789645644323719844389861950, 5.32760877515672001922052635416, 5.89659508378136399512481015178, 6.91898053873343207079042012278, 7.54378658277298552888846776515, 8.104953483061807478708330762148