L(s) = 1 | − 1.18·2-s + 2.30·3-s − 0.595·4-s + 3.24·5-s − 2.72·6-s − 2.76·7-s + 3.07·8-s + 2.30·9-s − 3.84·10-s + 5.58·11-s − 1.37·12-s + 1.64·13-s + 3.28·14-s + 7.46·15-s − 2.45·16-s − 0.758·17-s − 2.72·18-s + 2.17·19-s − 1.93·20-s − 6.37·21-s − 6.61·22-s − 0.00141·23-s + 7.08·24-s + 5.51·25-s − 1.94·26-s − 1.61·27-s + 1.64·28-s + ⋯ |
L(s) = 1 | − 0.838·2-s + 1.32·3-s − 0.297·4-s + 1.45·5-s − 1.11·6-s − 1.04·7-s + 1.08·8-s + 0.766·9-s − 1.21·10-s + 1.68·11-s − 0.395·12-s + 0.455·13-s + 0.876·14-s + 1.92·15-s − 0.613·16-s − 0.183·17-s − 0.642·18-s + 0.499·19-s − 0.431·20-s − 1.39·21-s − 1.41·22-s − 0.000294·23-s + 1.44·24-s + 1.10·25-s − 0.382·26-s − 0.309·27-s + 0.311·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.120588057\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.120588057\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 \) |
good | 2 | \( 1 + 1.18T + 2T^{2} \) |
| 3 | \( 1 - 2.30T + 3T^{2} \) |
| 5 | \( 1 - 3.24T + 5T^{2} \) |
| 7 | \( 1 + 2.76T + 7T^{2} \) |
| 11 | \( 1 - 5.58T + 11T^{2} \) |
| 13 | \( 1 - 1.64T + 13T^{2} \) |
| 17 | \( 1 + 0.758T + 17T^{2} \) |
| 19 | \( 1 - 2.17T + 19T^{2} \) |
| 23 | \( 1 + 0.00141T + 23T^{2} \) |
| 29 | \( 1 - 4.68T + 29T^{2} \) |
| 31 | \( 1 - 4.03T + 31T^{2} \) |
| 37 | \( 1 + 1.99T + 37T^{2} \) |
| 41 | \( 1 + 7.00T + 41T^{2} \) |
| 47 | \( 1 + 7.86T + 47T^{2} \) |
| 53 | \( 1 + 3.94T + 53T^{2} \) |
| 59 | \( 1 - 3.64T + 59T^{2} \) |
| 61 | \( 1 - 8.55T + 61T^{2} \) |
| 67 | \( 1 - 9.38T + 67T^{2} \) |
| 71 | \( 1 - 6.90T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 - 6.53T + 79T^{2} \) |
| 83 | \( 1 + 5.70T + 83T^{2} \) |
| 89 | \( 1 - 8.63T + 89T^{2} \) |
| 97 | \( 1 + 17.6T + 97T^{2} \) |
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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
−9.336573304865750289561814551503, −8.676320326962922533946875248488, −8.119066907395666639461108185842, −6.76787852319248455491743545385, −6.47385670812629217135287350877, −5.22672425418197926373951267202, −3.97470657287586115222477194670, −3.21357444296843078670409368836, −2.07098692501878898073120663100, −1.17241130452921591703815936557,
1.17241130452921591703815936557, 2.07098692501878898073120663100, 3.21357444296843078670409368836, 3.97470657287586115222477194670, 5.22672425418197926373951267202, 6.47385670812629217135287350877, 6.76787852319248455491743545385, 8.119066907395666639461108185842, 8.676320326962922533946875248488, 9.336573304865750289561814551503