L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 3.04·7-s + 8-s + 9-s + 10-s − 6.24·11-s − 12-s + 3.04·14-s − 15-s + 16-s + 2.69·17-s + 18-s − 5.82·19-s + 20-s − 3.04·21-s − 6.24·22-s − 5.62·23-s − 24-s + 25-s − 27-s + 3.04·28-s − 5.14·29-s − 30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 1.15·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 1.88·11-s − 0.288·12-s + 0.814·14-s − 0.258·15-s + 0.250·16-s + 0.652·17-s + 0.235·18-s − 1.33·19-s + 0.223·20-s − 0.665·21-s − 1.33·22-s − 1.17·23-s − 0.204·24-s + 0.200·25-s − 0.192·27-s + 0.576·28-s − 0.955·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 3.04T + 7T^{2} \) |
| 11 | \( 1 + 6.24T + 11T^{2} \) |
| 17 | \( 1 - 2.69T + 17T^{2} \) |
| 19 | \( 1 + 5.82T + 19T^{2} \) |
| 23 | \( 1 + 5.62T + 23T^{2} \) |
| 29 | \( 1 + 5.14T + 29T^{2} \) |
| 31 | \( 1 + 3.53T + 31T^{2} \) |
| 37 | \( 1 + 4.65T + 37T^{2} \) |
| 41 | \( 1 + 3.77T + 41T^{2} \) |
| 43 | \( 1 - 2.85T + 43T^{2} \) |
| 47 | \( 1 + 3.61T + 47T^{2} \) |
| 53 | \( 1 - 0.664T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 7.91T + 61T^{2} \) |
| 67 | \( 1 + 0.198T + 67T^{2} \) |
| 71 | \( 1 + 0.374T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 - 6.83T + 83T^{2} \) |
| 89 | \( 1 - 9.50T + 89T^{2} \) |
| 97 | \( 1 + 0.335T + 97T^{2} \) |
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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
−7.892623534880717559271282891636, −7.12992233146635180417079853408, −6.12036754823303647225339964998, −5.58752081022751678691594881929, −5.01171304331546331377700075806, −4.42429782397280536453648608514, −3.37483015173685662667362090383, −2.23891725154457608827322760922, −1.70467469228162191419275919954, 0,
1.70467469228162191419275919954, 2.23891725154457608827322760922, 3.37483015173685662667362090383, 4.42429782397280536453648608514, 5.01171304331546331377700075806, 5.58752081022751678691594881929, 6.12036754823303647225339964998, 7.12992233146635180417079853408, 7.892623534880717559271282891636