| L(s) = 1 | − 1.08·3-s + 4.41i·7-s − 7.83·9-s − 17.7·11-s + 15.5i·13-s − 10.8·17-s − 11.2·19-s − 4.77i·21-s − 25.2i·23-s + 18.1·27-s − 31.1i·29-s − 11.1i·31-s + 19.1·33-s + 57.5i·37-s − 16.8i·39-s + ⋯ |
| L(s) = 1 | − 0.360·3-s + 0.630i·7-s − 0.870·9-s − 1.61·11-s + 1.19i·13-s − 0.637·17-s − 0.592·19-s − 0.227i·21-s − 1.09i·23-s + 0.673·27-s − 1.07i·29-s − 0.360i·31-s + 0.580·33-s + 1.55i·37-s − 0.431i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7390826059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7390826059\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 1.08T + 9T^{2} \) |
| 7 | \( 1 - 4.41iT - 49T^{2} \) |
| 11 | \( 1 + 17.7T + 121T^{2} \) |
| 13 | \( 1 - 15.5iT - 169T^{2} \) |
| 17 | \( 1 + 10.8T + 289T^{2} \) |
| 19 | \( 1 + 11.2T + 361T^{2} \) |
| 23 | \( 1 + 25.2iT - 529T^{2} \) |
| 29 | \( 1 + 31.1iT - 841T^{2} \) |
| 31 | \( 1 + 11.1iT - 961T^{2} \) |
| 37 | \( 1 - 57.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 40.8T + 1.68e3T^{2} \) |
| 43 | \( 1 - 56.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 5.91iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 24.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 2.61T + 3.48e3T^{2} \) |
| 61 | \( 1 + 0.449iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 72.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 120. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 124.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 29.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 141.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 39.6T + 7.92e3T^{2} \) |
| 97 | \( 1 + 98.1T + 9.40e3T^{2} \) |
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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
−8.986097047387101051329891128415, −8.418325105661144279843735598389, −7.62512272706826844670290310847, −6.46335137655224674269414468359, −5.95586849704866630121180731866, −4.99378563196981459141247914261, −4.27767285508044663308326711230, −2.73494924143494817593276151669, −2.23407876816144975043726674273, −0.31162036669240480997794067462,
0.69149585962141296078986428667, 2.38407410597834841823876066557, 3.20882803349976889266597233148, 4.37636415227662046139648532490, 5.50393375082312445320645357557, 5.71049904871295039697847589275, 7.06635128140346812055595286967, 7.69833211302946785337279887015, 8.439620167137354004206996779340, 9.287321726142042353530459166610
