L(s) = 1 | + 2.18·2-s + 2.76·4-s − 0.541·5-s + 0.936·7-s + 1.66·8-s − 1.18·10-s + 3.69·11-s − 4.89·13-s + 2.04·14-s − 1.89·16-s − 7.10·17-s − 1.09·19-s − 1.49·20-s + 8.07·22-s − 23-s − 4.70·25-s − 10.6·26-s + 2.58·28-s − 29-s − 5.91·31-s − 7.45·32-s − 15.5·34-s − 0.507·35-s + 10.3·37-s − 2.39·38-s − 0.901·40-s + 0.971·41-s + ⋯ |
L(s) = 1 | + 1.54·2-s + 1.38·4-s − 0.242·5-s + 0.354·7-s + 0.588·8-s − 0.373·10-s + 1.11·11-s − 1.35·13-s + 0.546·14-s − 0.473·16-s − 1.72·17-s − 0.251·19-s − 0.334·20-s + 1.72·22-s − 0.208·23-s − 0.941·25-s − 2.09·26-s + 0.489·28-s − 0.185·29-s − 1.06·31-s − 1.31·32-s − 2.66·34-s − 0.0857·35-s + 1.70·37-s − 0.387·38-s − 0.142·40-s + 0.151·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 - 2.18T + 2T^{2} \) |
| 5 | \( 1 + 0.541T + 5T^{2} \) |
| 7 | \( 1 - 0.936T + 7T^{2} \) |
| 11 | \( 1 - 3.69T + 11T^{2} \) |
| 13 | \( 1 + 4.89T + 13T^{2} \) |
| 17 | \( 1 + 7.10T + 17T^{2} \) |
| 19 | \( 1 + 1.09T + 19T^{2} \) |
| 31 | \( 1 + 5.91T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 - 0.971T + 41T^{2} \) |
| 43 | \( 1 + 1.94T + 43T^{2} \) |
| 47 | \( 1 + 6.84T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 - 5.79T + 59T^{2} \) |
| 61 | \( 1 + 3.13T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 9.09T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 3.23T + 83T^{2} \) |
| 89 | \( 1 - 18.7T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−7.51344374001489388334513107898, −6.64592078890122849869415437016, −6.35837903759758448989470113012, −5.36639670306125531489942359247, −4.66560811299712672689345430087, −4.21624227811742171385830739693, −3.51193247406959109617730379283, −2.45499286284274137345609150101, −1.84335511985658204003812055761, 0,
1.84335511985658204003812055761, 2.45499286284274137345609150101, 3.51193247406959109617730379283, 4.21624227811742171385830739693, 4.66560811299712672689345430087, 5.36639670306125531489942359247, 6.35837903759758448989470113012, 6.64592078890122849869415437016, 7.51344374001489388334513107898