L(s) = 1 | − 0.539·3-s + 4.10·5-s − 7-s − 2.70·9-s − 2.76·11-s − 4.05·13-s − 2.21·15-s + 5.22·17-s + 0.109·19-s + 0.539·21-s − 6.08·23-s + 11.8·25-s + 3.08·27-s + 2.25·29-s + 1.18·31-s + 1.49·33-s − 4.10·35-s − 8.95·37-s + 2.18·39-s − 41-s + 7.93·43-s − 11.1·45-s + 10.5·47-s + 49-s − 2.81·51-s + 6.23·53-s − 11.3·55-s + ⋯ |
L(s) = 1 | − 0.311·3-s + 1.83·5-s − 0.377·7-s − 0.902·9-s − 0.834·11-s − 1.12·13-s − 0.571·15-s + 1.26·17-s + 0.0250·19-s + 0.117·21-s − 1.26·23-s + 2.36·25-s + 0.592·27-s + 0.418·29-s + 0.212·31-s + 0.260·33-s − 0.693·35-s − 1.47·37-s + 0.350·39-s − 0.156·41-s + 1.21·43-s − 1.65·45-s + 1.54·47-s + 0.142·49-s − 0.394·51-s + 0.855·53-s − 1.53·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.917841580\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.917841580\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 + 0.539T + 3T^{2} \) |
| 5 | \( 1 - 4.10T + 5T^{2} \) |
| 11 | \( 1 + 2.76T + 11T^{2} \) |
| 13 | \( 1 + 4.05T + 13T^{2} \) |
| 17 | \( 1 - 5.22T + 17T^{2} \) |
| 19 | \( 1 - 0.109T + 19T^{2} \) |
| 23 | \( 1 + 6.08T + 23T^{2} \) |
| 29 | \( 1 - 2.25T + 29T^{2} \) |
| 31 | \( 1 - 1.18T + 31T^{2} \) |
| 37 | \( 1 + 8.95T + 37T^{2} \) |
| 43 | \( 1 - 7.93T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 6.23T + 53T^{2} \) |
| 59 | \( 1 - 9.43T + 59T^{2} \) |
| 61 | \( 1 - 6.89T + 61T^{2} \) |
| 67 | \( 1 - 1.35T + 67T^{2} \) |
| 71 | \( 1 - 7.90T + 71T^{2} \) |
| 73 | \( 1 - 9.88T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 + 0.852T + 89T^{2} \) |
| 97 | \( 1 - 9.92T + 97T^{2} \) |
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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
−8.367053808429108865545553902196, −7.57161015582169034402493197504, −6.66814269352914409146910915293, −6.00900547159851103059615463457, −5.30785121004310912842688439656, −5.16871614348470341787158180926, −3.63988028853190716488082208153, −2.52356896758799149308776611265, −2.24645461374067055315628909247, −0.75819213595598159214198956946,
0.75819213595598159214198956946, 2.24645461374067055315628909247, 2.52356896758799149308776611265, 3.63988028853190716488082208153, 5.16871614348470341787158180926, 5.30785121004310912842688439656, 6.00900547159851103059615463457, 6.66814269352914409146910915293, 7.57161015582169034402493197504, 8.367053808429108865545553902196