| L(s) = 1 | − 4.35i·5-s − 13.0·7-s − 9·9-s + 3i·11-s + 15·17-s − 19i·19-s − 34.8·23-s + 5.99·25-s + 57.0i·35-s + 85i·43-s + 39.2i·45-s + 56.6·47-s + 122·49-s + 13.0·55-s + 65.3i·61-s + ⋯ |
| L(s) = 1 | − 0.871i·5-s − 1.86·7-s − 9-s + 0.272i·11-s + 0.882·17-s − i·19-s − 1.51·23-s + 0.239·25-s + 1.62i·35-s + 1.97i·43-s + 0.871i·45-s + 1.20·47-s + 2.48·49-s + 0.237·55-s + 1.07i·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7611991974\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7611991974\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 19iT \) |
| good | 3 | \( 1 + 9T^{2} \) |
| 5 | \( 1 + 4.35iT - 25T^{2} \) |
| 7 | \( 1 + 13.0T + 49T^{2} \) |
| 11 | \( 1 - 3iT - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 - 15T + 289T^{2} \) |
| 23 | \( 1 + 34.8T + 529T^{2} \) |
| 29 | \( 1 + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 85iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 56.6T + 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 + 3.48e3T^{2} \) |
| 61 | \( 1 - 65.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 25T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 90iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−9.568768006447120731664298314099, −8.964960560586896922541387824459, −8.145116376610775568476398541555, −7.10886532713879109911673091627, −6.16168478765849599644157746305, −5.60715276406993970333564183459, −4.45038270462622015071321158910, −3.37572316841423710882084213483, −2.55242593638657475084624966556, −0.76911522140546938592787444503,
0.31779250022731534242727266464, 2.36095500245126074794135279123, 3.29674438510883555350660116027, 3.76421505406553121119463427551, 5.60754123219242327783496731016, 6.06691273821743433232399550192, 6.82177817367526266328973466098, 7.73825880551706435988489071730, 8.689350981507256531863976946174, 9.578141505504127742883183024232
