L(s) = 1 | + 2-s − 3.23·3-s + 4-s − 1.38·5-s − 3.23·6-s + 1.23·7-s + 8-s + 7.47·9-s − 1.38·10-s + 1.23·11-s − 3.23·12-s − 3.61·13-s + 1.23·14-s + 4.47·15-s + 16-s − 5.61·17-s + 7.47·18-s − 1.38·20-s − 4.00·21-s + 1.23·22-s + 0.763·23-s − 3.23·24-s − 3.09·25-s − 3.61·26-s − 14.4·27-s + 1.23·28-s − 2.09·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.86·3-s + 0.5·4-s − 0.618·5-s − 1.32·6-s + 0.467·7-s + 0.353·8-s + 2.49·9-s − 0.437·10-s + 0.372·11-s − 0.934·12-s − 1.00·13-s + 0.330·14-s + 1.15·15-s + 0.250·16-s − 1.36·17-s + 1.76·18-s − 0.309·20-s − 0.872·21-s + 0.263·22-s + 0.159·23-s − 0.660·24-s − 0.618·25-s − 0.709·26-s − 2.78·27-s + 0.233·28-s − 0.388·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + 3.23T + 3T^{2} \) |
| 5 | \( 1 + 1.38T + 5T^{2} \) |
| 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 - 1.23T + 11T^{2} \) |
| 13 | \( 1 + 3.61T + 13T^{2} \) |
| 17 | \( 1 + 5.61T + 17T^{2} \) |
| 23 | \( 1 - 0.763T + 23T^{2} \) |
| 29 | \( 1 + 2.09T + 29T^{2} \) |
| 31 | \( 1 + 3.23T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 5.85T + 41T^{2} \) |
| 43 | \( 1 + 4.76T + 43T^{2} \) |
| 47 | \( 1 + 4.47T + 47T^{2} \) |
| 53 | \( 1 + 5.09T + 53T^{2} \) |
| 59 | \( 1 + 8.47T + 59T^{2} \) |
| 61 | \( 1 - 3.61T + 61T^{2} \) |
| 67 | \( 1 - 1.70T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 + 3.38T + 73T^{2} \) |
| 79 | \( 1 - 7.23T + 79T^{2} \) |
| 83 | \( 1 - 8.47T + 83T^{2} \) |
| 89 | \( 1 + 2.14T + 89T^{2} \) |
| 97 | \( 1 + 6.38T + 97T^{2} \) |
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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
−10.41917633871011344614954107809, −9.366197516328403205041091650127, −7.85338920246320079757324961277, −6.98320448599618576482942643405, −6.37493376386017553899124820146, −5.25189637084603235053397294873, −4.73054734024685847293227851885, −3.83298704071778634887001984899, −1.83733823599422545968857624565, 0,
1.83733823599422545968857624565, 3.83298704071778634887001984899, 4.73054734024685847293227851885, 5.25189637084603235053397294873, 6.37493376386017553899124820146, 6.98320448599618576482942643405, 7.85338920246320079757324961277, 9.366197516328403205041091650127, 10.41917633871011344614954107809