L(s) = 1 | − 0.849·2-s − 3-s − 1.27·4-s − 5-s + 0.849·6-s − 1.45·7-s + 2.78·8-s + 9-s + 0.849·10-s + 3.66·11-s + 1.27·12-s − 13-s + 1.23·14-s + 15-s + 0.191·16-s − 6.44·17-s − 0.849·18-s + 0.290·19-s + 1.27·20-s + 1.45·21-s − 3.11·22-s + 3.97·23-s − 2.78·24-s + 25-s + 0.849·26-s − 27-s + 1.86·28-s + ⋯ |
L(s) = 1 | − 0.600·2-s − 0.577·3-s − 0.639·4-s − 0.447·5-s + 0.346·6-s − 0.551·7-s + 0.984·8-s + 0.333·9-s + 0.268·10-s + 1.10·11-s + 0.369·12-s − 0.277·13-s + 0.331·14-s + 0.258·15-s + 0.0479·16-s − 1.56·17-s − 0.200·18-s + 0.0665·19-s + 0.285·20-s + 0.318·21-s − 0.663·22-s + 0.829·23-s − 0.568·24-s + 0.200·25-s + 0.166·26-s − 0.192·27-s + 0.352·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
good | 2 | \( 1 + 0.849T + 2T^{2} \) |
| 7 | \( 1 + 1.45T + 7T^{2} \) |
| 11 | \( 1 - 3.66T + 11T^{2} \) |
| 17 | \( 1 + 6.44T + 17T^{2} \) |
| 19 | \( 1 - 0.290T + 19T^{2} \) |
| 23 | \( 1 - 3.97T + 23T^{2} \) |
| 29 | \( 1 + 7.99T + 29T^{2} \) |
| 37 | \( 1 + 2.74T + 37T^{2} \) |
| 41 | \( 1 - 8.38T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 2.98T + 47T^{2} \) |
| 53 | \( 1 - 8.28T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 6.31T + 61T^{2} \) |
| 67 | \( 1 - 8.50T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 - 1.21T + 73T^{2} \) |
| 79 | \( 1 + 6.05T + 79T^{2} \) |
| 83 | \( 1 - 8.18T + 83T^{2} \) |
| 89 | \( 1 + 0.797T + 89T^{2} \) |
| 97 | \( 1 - 3.18T + 97T^{2} \) |
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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
−7.53183132069310001167088863491, −7.22567699827104355561722861011, −6.36652938403013811180148877015, −5.65769636892684670304505514880, −4.62256301986015853441664579411, −4.22027638242248321796558620424, −3.40690225337465415041424226724, −2.08003138169264955860352027862, −0.949528090908707460591922244822, 0,
0.949528090908707460591922244822, 2.08003138169264955860352027862, 3.40690225337465415041424226724, 4.22027638242248321796558620424, 4.62256301986015853441664579411, 5.65769636892684670304505514880, 6.36652938403013811180148877015, 7.22567699827104355561722861011, 7.53183132069310001167088863491