L(s) = 1 | − 2-s + 3-s + 4-s − 1.18·5-s − 6-s + 0.107·7-s − 8-s + 9-s + 1.18·10-s + 0.456·11-s + 12-s + 13-s − 0.107·14-s − 1.18·15-s + 16-s − 3.46·17-s − 18-s − 0.368·19-s − 1.18·20-s + 0.107·21-s − 0.456·22-s + 5.15·23-s − 24-s − 3.58·25-s − 26-s + 27-s + 0.107·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.531·5-s − 0.408·6-s + 0.0404·7-s − 0.353·8-s + 0.333·9-s + 0.375·10-s + 0.137·11-s + 0.288·12-s + 0.277·13-s − 0.0286·14-s − 0.306·15-s + 0.250·16-s − 0.841·17-s − 0.235·18-s − 0.0846·19-s − 0.265·20-s + 0.0233·21-s − 0.0973·22-s + 1.07·23-s − 0.204·24-s − 0.717·25-s − 0.196·26-s + 0.192·27-s + 0.0202·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 5 | \( 1 + 1.18T + 5T^{2} \) |
| 7 | \( 1 - 0.107T + 7T^{2} \) |
| 11 | \( 1 - 0.456T + 11T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 0.368T + 19T^{2} \) |
| 23 | \( 1 - 5.15T + 23T^{2} \) |
| 29 | \( 1 + 3.44T + 29T^{2} \) |
| 31 | \( 1 - 0.0661T + 31T^{2} \) |
| 37 | \( 1 - 4.66T + 37T^{2} \) |
| 41 | \( 1 - 0.724T + 41T^{2} \) |
| 43 | \( 1 - 3.85T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 + 7.67T + 53T^{2} \) |
| 59 | \( 1 - 4.84T + 59T^{2} \) |
| 61 | \( 1 - 6.69T + 61T^{2} \) |
| 67 | \( 1 - 2.46T + 67T^{2} \) |
| 71 | \( 1 - 6.86T + 71T^{2} \) |
| 73 | \( 1 + 7.04T + 73T^{2} \) |
| 79 | \( 1 - 0.265T + 79T^{2} \) |
| 83 | \( 1 - 0.324T + 83T^{2} \) |
| 89 | \( 1 + 4.77T + 89T^{2} \) |
| 97 | \( 1 - 3.46T + 97T^{2} \) |
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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
−7.72765948043621793360188053133, −6.88863973628612146483436493902, −6.42830784762656074955815774977, −5.42466092839219733326348445762, −4.52157709869060984406451134236, −3.78476675282600859085506166686, −3.02087730712551374560982209780, −2.16069953989833725867600471252, −1.25124801471604180707492844182, 0,
1.25124801471604180707492844182, 2.16069953989833725867600471252, 3.02087730712551374560982209780, 3.78476675282600859085506166686, 4.52157709869060984406451134236, 5.42466092839219733326348445762, 6.42830784762656074955815774977, 6.88863973628612146483436493902, 7.72765948043621793360188053133