L(s) = 1 | − 2.16·2-s + 3-s + 2.68·4-s + 5-s − 2.16·6-s − 0.480·7-s − 1.48·8-s + 9-s − 2.16·10-s + 2.68·12-s − 3.36·13-s + 1.03·14-s + 15-s − 2.16·16-s − 4.84·17-s − 2.16·18-s + 1.84·19-s + 2.68·20-s − 0.480·21-s + 3.48·23-s − 1.48·24-s + 25-s + 7.28·26-s + 27-s − 1.28·28-s + 8.32·29-s − 2.16·30-s + ⋯ |
L(s) = 1 | − 1.53·2-s + 0.577·3-s + 1.34·4-s + 0.447·5-s − 0.883·6-s − 0.181·7-s − 0.523·8-s + 0.333·9-s − 0.684·10-s + 0.774·12-s − 0.934·13-s + 0.277·14-s + 0.258·15-s − 0.541·16-s − 1.17·17-s − 0.510·18-s + 0.424·19-s + 0.600·20-s − 0.104·21-s + 0.725·23-s − 0.302·24-s + 0.200·25-s + 1.42·26-s + 0.192·27-s − 0.243·28-s + 1.54·29-s − 0.395·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9793963044\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9793963044\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 2.16T + 2T^{2} \) |
| 7 | \( 1 + 0.480T + 7T^{2} \) |
| 13 | \( 1 + 3.36T + 13T^{2} \) |
| 17 | \( 1 + 4.84T + 17T^{2} \) |
| 19 | \( 1 - 1.84T + 19T^{2} \) |
| 23 | \( 1 - 3.48T + 23T^{2} \) |
| 29 | \( 1 - 8.32T + 29T^{2} \) |
| 31 | \( 1 + 2.36T + 31T^{2} \) |
| 37 | \( 1 - 5.44T + 37T^{2} \) |
| 41 | \( 1 - 6.65T + 41T^{2} \) |
| 43 | \( 1 + 4.32T + 43T^{2} \) |
| 47 | \( 1 + 3.17T + 47T^{2} \) |
| 53 | \( 1 - 13.5T + 53T^{2} \) |
| 59 | \( 1 - 15.0T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 8.17T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + 5.84T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 5.67T + 89T^{2} \) |
| 97 | \( 1 + 8.17T + 97T^{2} \) |
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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
−9.275465343018423949041030445203, −8.600454977200911188984058013287, −7.958971128028535731626569660208, −7.00883826678927327401722549710, −6.63704595542074971390252223261, −5.24225845762575283357219830946, −4.26611802124737412364292139433, −2.78516966713804589739700595851, −2.13136431246681253331907338546, −0.824133369501425312176330807248,
0.824133369501425312176330807248, 2.13136431246681253331907338546, 2.78516966713804589739700595851, 4.26611802124737412364292139433, 5.24225845762575283357219830946, 6.63704595542074971390252223261, 7.00883826678927327401722549710, 7.958971128028535731626569660208, 8.600454977200911188984058013287, 9.275465343018423949041030445203