L(s) = 1 | + 5.45·2-s + 6.71·3-s + 21.7·4-s + 3.97·5-s + 36.6·6-s + 10.9·7-s + 74.8·8-s + 18.0·9-s + 21.6·10-s − 11·11-s + 145.·12-s + 59.8·14-s + 26.7·15-s + 234.·16-s + 133.·17-s + 98.6·18-s − 49.7·19-s + 86.3·20-s + 73.7·21-s − 59.9·22-s − 148.·23-s + 502.·24-s − 109.·25-s − 59.7·27-s + 238.·28-s + 51.1·29-s + 145.·30-s + ⋯ |
L(s) = 1 | + 1.92·2-s + 1.29·3-s + 2.71·4-s + 0.355·5-s + 2.49·6-s + 0.592·7-s + 3.30·8-s + 0.670·9-s + 0.685·10-s − 0.301·11-s + 3.50·12-s + 1.14·14-s + 0.459·15-s + 3.65·16-s + 1.90·17-s + 1.29·18-s − 0.600·19-s + 0.965·20-s + 0.766·21-s − 0.581·22-s − 1.34·23-s + 4.27·24-s − 0.873·25-s − 0.426·27-s + 1.60·28-s + 0.327·29-s + 0.886·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(15.27257278\) |
\(L(\frac12)\) |
\(\approx\) |
\(15.27257278\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 5.45T + 8T^{2} \) |
| 3 | \( 1 - 6.71T + 27T^{2} \) |
| 5 | \( 1 - 3.97T + 125T^{2} \) |
| 7 | \( 1 - 10.9T + 343T^{2} \) |
| 17 | \( 1 - 133.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 49.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 148.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 51.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 39.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 65.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 169.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 371.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 263.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 327.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 401.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 219.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 514.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 86.9T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.13e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 238.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 474.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 58.0T + 7.04e5T^{2} \) |
| 97 | \( 1 - 982.T + 9.12e5T^{2} \) |
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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
−8.556800578681303171601546330183, −7.86531284464011550389094865867, −7.33439632523437594584600835597, −6.13864195271828982226674150452, −5.57461341318957582579455755509, −4.67080236411411233002682827332, −3.76302931719547624690646275585, −3.18017099583540090129281010507, −2.24046188120018238678467558808, −1.60647557901984227046368310067,
1.60647557901984227046368310067, 2.24046188120018238678467558808, 3.18017099583540090129281010507, 3.76302931719547624690646275585, 4.67080236411411233002682827332, 5.57461341318957582579455755509, 6.13864195271828982226674150452, 7.33439632523437594584600835597, 7.86531284464011550389094865867, 8.556800578681303171601546330183