| L(s) = 1 | + 2-s + 3.01·3-s + 4-s − 2.00·5-s + 3.01·6-s + 2.42·7-s + 8-s + 6.08·9-s − 2.00·10-s + 3.01·12-s + 3.03·13-s + 2.42·14-s − 6.05·15-s + 16-s − 4.31·17-s + 6.08·18-s − 19-s − 2.00·20-s + 7.32·21-s − 2.27·23-s + 3.01·24-s − 0.960·25-s + 3.03·26-s + 9.30·27-s + 2.42·28-s + 8.81·29-s − 6.05·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.74·3-s + 0.5·4-s − 0.898·5-s + 1.23·6-s + 0.917·7-s + 0.353·8-s + 2.02·9-s − 0.635·10-s + 0.870·12-s + 0.841·13-s + 0.649·14-s − 1.56·15-s + 0.250·16-s − 1.04·17-s + 1.43·18-s − 0.229·19-s − 0.449·20-s + 1.59·21-s − 0.475·23-s + 0.615·24-s − 0.192·25-s + 0.595·26-s + 1.79·27-s + 0.458·28-s + 1.63·29-s − 1.10·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.943420864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.943420864\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 3.01T + 3T^{2} \) |
| 5 | \( 1 + 2.00T + 5T^{2} \) |
| 7 | \( 1 - 2.42T + 7T^{2} \) |
| 13 | \( 1 - 3.03T + 13T^{2} \) |
| 17 | \( 1 + 4.31T + 17T^{2} \) |
| 23 | \( 1 + 2.27T + 23T^{2} \) |
| 29 | \( 1 - 8.81T + 29T^{2} \) |
| 31 | \( 1 - 9.29T + 31T^{2} \) |
| 37 | \( 1 - 1.06T + 37T^{2} \) |
| 41 | \( 1 + 5.15T + 41T^{2} \) |
| 43 | \( 1 - 11.9T + 43T^{2} \) |
| 47 | \( 1 + 9.00T + 47T^{2} \) |
| 53 | \( 1 - 5.09T + 53T^{2} \) |
| 59 | \( 1 + 4.79T + 59T^{2} \) |
| 61 | \( 1 - 9.26T + 61T^{2} \) |
| 67 | \( 1 - 9.19T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 8.93T + 73T^{2} \) |
| 79 | \( 1 - 6.52T + 79T^{2} \) |
| 83 | \( 1 + 4.09T + 83T^{2} \) |
| 89 | \( 1 - 6.04T + 89T^{2} \) |
| 97 | \( 1 + 9.42T + 97T^{2} \) |
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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
−8.129573974675886913830384368675, −7.992238453080517729671627145081, −6.93977922763763840557724214263, −6.34402621930353220810884696346, −4.99587261525464833915413353592, −4.22472709748023381537523683976, −3.93529574935267888904175509506, −2.92590346686923999020642995838, −2.28045417514315707256469793049, −1.25109273387972170750529827206,
1.25109273387972170750529827206, 2.28045417514315707256469793049, 2.92590346686923999020642995838, 3.93529574935267888904175509506, 4.22472709748023381537523683976, 4.99587261525464833915413353592, 6.34402621930353220810884696346, 6.93977922763763840557724214263, 7.992238453080517729671627145081, 8.129573974675886913830384368675
