| L(s) = 1 | + (1.28 + 0.581i)2-s + (1.32 + 1.50i)4-s + 2.64·7-s + (0.832 + 2.70i)8-s − 0.913·11-s + (3.41 + 1.53i)14-s + (−0.5 + 3.96i)16-s + (−1.17 − 0.531i)22-s − 1.91i·23-s + 5·25-s + (3.50 + 3.96i)28-s − 6.06·29-s + (−2.95 + 4.82i)32-s + 6i·37-s − 12i·43-s + (−1.20 − 1.36i)44-s + ⋯ |
| L(s) = 1 | + (0.911 + 0.411i)2-s + (0.661 + 0.750i)4-s + 0.999·7-s + (0.294 + 0.955i)8-s − 0.275·11-s + (0.911 + 0.411i)14-s + (−0.125 + 0.992i)16-s + (−0.250 − 0.113i)22-s − 0.399i·23-s + 25-s + (0.661 + 0.749i)28-s − 1.12·29-s + (−0.522 + 0.852i)32-s + 0.986i·37-s − 1.82i·43-s + (−0.182 − 0.206i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 - 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.610 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.34999 + 1.15580i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.34999 + 1.15580i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.28 - 0.581i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 2.64T \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 11 | \( 1 + 0.913T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 1.91iT - 23T^{2} \) |
| 29 | \( 1 + 6.06T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 12iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 14.5T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 15.8iT - 67T^{2} \) |
| 71 | \( 1 + 15.0iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−11.16127153415635245221039430503, −10.51695568284184444015066616197, −9.027881753140462061127684428858, −8.119195383287518434686997903471, −7.37455531581442569529636552201, −6.34569303945833506956233700797, −5.25102696016211461924201313620, −4.56794876030894144225384640623, −3.31183434682759315256083406204, −1.94685076960933944687616041987,
1.48299204425161403449017861510, 2.78748919046481694332847465141, 4.09528387151470492255306197354, 5.01660455476718319845645002886, 5.84742941592529932442124266330, 7.05561167736618777777129457270, 7.934269719259214630383264492196, 9.146316812769200140578727525141, 10.17349841648640844247658475425, 11.13582159668544337735927165484
