| L(s) = 1 | − i·3-s + (−2 − i)5-s + i·7-s + 2·9-s − 11-s − i·13-s + (−1 + 2i)15-s − 3i·17-s + 4·19-s + 21-s + 2i·23-s + (3 + 4i)25-s − 5i·27-s − 29-s + 6·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (−0.894 − 0.447i)5-s + 0.377i·7-s + 0.666·9-s − 0.301·11-s − 0.277i·13-s + (−0.258 + 0.516i)15-s − 0.727i·17-s + 0.917·19-s + 0.218·21-s + 0.417i·23-s + (0.600 + 0.800i)25-s − 0.962i·27-s − 0.185·29-s + 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.247734096\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.247734096\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 + iT - 3T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 9iT - 47T^{2} \) |
| 53 | \( 1 + 14iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 11T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 + 19iT - 97T^{2} \) |
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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
−8.542424547030526254761347178488, −8.025460007028469154778905167366, −7.24417849806341420839282060698, −6.73101850042924224945473979757, −5.46269164017415978342635047579, −4.88436179121503404212263908218, −3.84390421675141423186622724740, −2.93048892982389089818352807581, −1.65999599518093312425907452823, −0.48688561812222967842279542421,
1.24113238503267741471942582820, 2.79199601537814098102097314574, 3.68700826587071253205425586498, 4.35083301824026702807948461899, 5.07564597879093575582859771829, 6.31319940054283498488125816837, 7.02426268440400503282985943107, 7.76250499372321739481278893558, 8.393090239070038574813464941167, 9.420317614107810252822301409860
