L(s) = 1 | + 2-s − 0.676·3-s + 4-s − 1.65·5-s − 0.676·6-s + 3.29·7-s + 8-s − 2.54·9-s − 1.65·10-s + 5.58·11-s − 0.676·12-s − 0.428·13-s + 3.29·14-s + 1.11·15-s + 16-s + 3.61·17-s − 2.54·18-s + 3.36·19-s − 1.65·20-s − 2.23·21-s + 5.58·22-s + 0.496·23-s − 0.676·24-s − 2.26·25-s − 0.428·26-s + 3.74·27-s + 3.29·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.390·3-s + 0.5·4-s − 0.739·5-s − 0.276·6-s + 1.24·7-s + 0.353·8-s − 0.847·9-s − 0.523·10-s + 1.68·11-s − 0.195·12-s − 0.118·13-s + 0.881·14-s + 0.289·15-s + 0.250·16-s + 0.875·17-s − 0.599·18-s + 0.772·19-s − 0.369·20-s − 0.487·21-s + 1.19·22-s + 0.103·23-s − 0.138·24-s − 0.452·25-s − 0.0839·26-s + 0.721·27-s + 0.623·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.879030163\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.879030163\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 2003 | \( 1 + T \) |
good | 3 | \( 1 + 0.676T + 3T^{2} \) |
| 5 | \( 1 + 1.65T + 5T^{2} \) |
| 7 | \( 1 - 3.29T + 7T^{2} \) |
| 11 | \( 1 - 5.58T + 11T^{2} \) |
| 13 | \( 1 + 0.428T + 13T^{2} \) |
| 17 | \( 1 - 3.61T + 17T^{2} \) |
| 19 | \( 1 - 3.36T + 19T^{2} \) |
| 23 | \( 1 - 0.496T + 23T^{2} \) |
| 29 | \( 1 + 2.59T + 29T^{2} \) |
| 31 | \( 1 + 8.75T + 31T^{2} \) |
| 37 | \( 1 - 1.22T + 37T^{2} \) |
| 41 | \( 1 - 3.45T + 41T^{2} \) |
| 43 | \( 1 - 9.11T + 43T^{2} \) |
| 47 | \( 1 + 2.38T + 47T^{2} \) |
| 53 | \( 1 - 3.55T + 53T^{2} \) |
| 59 | \( 1 - 6.55T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 - 0.840T + 67T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 - 2.78T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 9.96T + 83T^{2} \) |
| 89 | \( 1 - 1.84T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−8.368043647630027451459297757754, −7.50244800001510126403051003592, −7.11790503502283514722106245464, −5.88186888862666576581883786636, −5.58585909590680858442705460826, −4.61415995525263253415827168370, −3.93819129795867708132344821521, −3.25005751365221003888333620971, −1.94904456621015245371700198214, −0.943678577942730376577385714851,
0.943678577942730376577385714851, 1.94904456621015245371700198214, 3.25005751365221003888333620971, 3.93819129795867708132344821521, 4.61415995525263253415827168370, 5.58585909590680858442705460826, 5.88186888862666576581883786636, 7.11790503502283514722106245464, 7.50244800001510126403051003592, 8.368043647630027451459297757754