L(s) = 1 | − 1.24·2-s − 0.445·4-s + 2.80·5-s − 4.80·7-s + 3.04·8-s − 3.49·10-s − 1.46·11-s + 5.98·14-s − 2.91·16-s + 2.44·17-s + 2.54·19-s − 1.24·20-s + 1.82·22-s + 3.51·23-s + 2.85·25-s + 2.13·28-s − 1.85·29-s − 7.63·31-s − 2.46·32-s − 3.04·34-s − 13.4·35-s − 4.55·37-s − 3.17·38-s + 8.54·40-s − 1.24·41-s + 2.38·43-s + 0.652·44-s + ⋯ |
L(s) = 1 | − 0.881·2-s − 0.222·4-s + 1.25·5-s − 1.81·7-s + 1.07·8-s − 1.10·10-s − 0.442·11-s + 1.60·14-s − 0.727·16-s + 0.593·17-s + 0.583·19-s − 0.278·20-s + 0.389·22-s + 0.733·23-s + 0.570·25-s + 0.403·28-s − 0.343·29-s − 1.37·31-s − 0.436·32-s − 0.522·34-s − 2.27·35-s − 0.748·37-s − 0.514·38-s + 1.35·40-s − 0.194·41-s + 0.363·43-s + 0.0984·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 1.24T + 2T^{2} \) |
| 5 | \( 1 - 2.80T + 5T^{2} \) |
| 7 | \( 1 + 4.80T + 7T^{2} \) |
| 11 | \( 1 + 1.46T + 11T^{2} \) |
| 17 | \( 1 - 2.44T + 17T^{2} \) |
| 19 | \( 1 - 2.54T + 19T^{2} \) |
| 23 | \( 1 - 3.51T + 23T^{2} \) |
| 29 | \( 1 + 1.85T + 29T^{2} \) |
| 31 | \( 1 + 7.63T + 31T^{2} \) |
| 37 | \( 1 + 4.55T + 37T^{2} \) |
| 41 | \( 1 + 1.24T + 41T^{2} \) |
| 43 | \( 1 - 2.38T + 43T^{2} \) |
| 47 | \( 1 + 12.8T + 47T^{2} \) |
| 53 | \( 1 - 8.85T + 53T^{2} \) |
| 59 | \( 1 + 2.17T + 59T^{2} \) |
| 61 | \( 1 + 7.82T + 61T^{2} \) |
| 67 | \( 1 + 3.58T + 67T^{2} \) |
| 71 | \( 1 - 8.83T + 71T^{2} \) |
| 73 | \( 1 + 7.69T + 73T^{2} \) |
| 79 | \( 1 + 4.02T + 79T^{2} \) |
| 83 | \( 1 - 0.652T + 83T^{2} \) |
| 89 | \( 1 + 6.29T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−9.427112669523287395114987350414, −8.563137452600505583038366475641, −7.42971359446796852427066526973, −6.77676663227556136629680344727, −5.79726859550645205918331966796, −5.18318281863811362304794312606, −3.72021861383773522017958466972, −2.78142834641264025903654852269, −1.48025328363071115284434145052, 0,
1.48025328363071115284434145052, 2.78142834641264025903654852269, 3.72021861383773522017958466972, 5.18318281863811362304794312606, 5.79726859550645205918331966796, 6.77676663227556136629680344727, 7.42971359446796852427066526973, 8.563137452600505583038366475641, 9.427112669523287395114987350414