| L(s) = 1 | + 1.90·2-s + 1.42·3-s + 1.64·4-s + 5-s + 2.71·6-s − 2.00·7-s − 0.682·8-s − 0.982·9-s + 1.90·10-s − 1.97·11-s + 2.33·12-s + 1.21·13-s − 3.83·14-s + 1.42·15-s − 4.58·16-s − 17-s − 1.87·18-s − 1.68·19-s + 1.64·20-s − 2.85·21-s − 3.76·22-s + 1.70·23-s − 0.969·24-s + 25-s + 2.31·26-s − 5.65·27-s − 3.29·28-s + ⋯ |
| L(s) = 1 | + 1.34·2-s + 0.820·3-s + 0.821·4-s + 0.447·5-s + 1.10·6-s − 0.758·7-s − 0.241·8-s − 0.327·9-s + 0.603·10-s − 0.594·11-s + 0.673·12-s + 0.335·13-s − 1.02·14-s + 0.366·15-s − 1.14·16-s − 0.242·17-s − 0.441·18-s − 0.385·19-s + 0.367·20-s − 0.622·21-s − 0.802·22-s + 0.355·23-s − 0.197·24-s + 0.200·25-s + 0.453·26-s − 1.08·27-s − 0.623·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| good | 2 | \( 1 - 1.90T + 2T^{2} \) |
| 3 | \( 1 - 1.42T + 3T^{2} \) |
| 7 | \( 1 + 2.00T + 7T^{2} \) |
| 11 | \( 1 + 1.97T + 11T^{2} \) |
| 13 | \( 1 - 1.21T + 13T^{2} \) |
| 19 | \( 1 + 1.68T + 19T^{2} \) |
| 23 | \( 1 - 1.70T + 23T^{2} \) |
| 29 | \( 1 + 8.70T + 29T^{2} \) |
| 31 | \( 1 - 7.83T + 31T^{2} \) |
| 37 | \( 1 - 0.948T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 - 2.19T + 43T^{2} \) |
| 47 | \( 1 + 1.75T + 47T^{2} \) |
| 53 | \( 1 - 1.74T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 - 8.29T + 61T^{2} \) |
| 67 | \( 1 + 9.76T + 67T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 + 0.936T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 + 0.0347T + 89T^{2} \) |
| 97 | \( 1 + 8.37T + 97T^{2} \) |
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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
−7.64839429098484575699414565465, −6.74059030359591243674280568706, −6.13995048674493377999969072095, −5.54947996199451492361023036309, −4.79390499829469160617988256471, −3.93428773623833717480579318105, −3.19539576399459881419581367806, −2.75353613215280261794505098935, −1.87303910172863501295352188177, 0,
1.87303910172863501295352188177, 2.75353613215280261794505098935, 3.19539576399459881419581367806, 3.93428773623833717480579318105, 4.79390499829469160617988256471, 5.54947996199451492361023036309, 6.13995048674493377999969072095, 6.74059030359591243674280568706, 7.64839429098484575699414565465
