| L(s) = 1 | − i·3-s − 6i·7-s + 26·9-s − 19·11-s − 12i·13-s − 75i·17-s + 91·19-s − 6·21-s − 174i·23-s − 53i·27-s + 272·29-s − 230·31-s + 19i·33-s − 182i·37-s − 12·39-s + ⋯ |
| L(s) = 1 | − 0.192i·3-s − 0.323i·7-s + 0.962·9-s − 0.520·11-s − 0.256i·13-s − 1.07i·17-s + 1.09·19-s − 0.0623·21-s − 1.57i·23-s − 0.377i·27-s + 1.74·29-s − 1.33·31-s + 0.100i·33-s − 0.808i·37-s − 0.0492·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.45735 - 0.900694i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.45735 - 0.900694i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + iT - 27T^{2} \) |
| 7 | \( 1 + 6iT - 343T^{2} \) |
| 11 | \( 1 + 19T + 1.33e3T^{2} \) |
| 13 | \( 1 + 12iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 75iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 91T + 6.85e3T^{2} \) |
| 23 | \( 1 + 174iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 272T + 2.43e4T^{2} \) |
| 31 | \( 1 + 230T + 2.97e4T^{2} \) |
| 37 | \( 1 + 182iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 117T + 6.89e4T^{2} \) |
| 43 | \( 1 + 372iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 52iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 402iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 312T + 2.05e5T^{2} \) |
| 61 | \( 1 - 170T + 2.26e5T^{2} \) |
| 67 | \( 1 - 763iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 52T + 3.57e5T^{2} \) |
| 73 | \( 1 - 981iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.05e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 351iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 799T + 7.04e5T^{2} \) |
| 97 | \( 1 - 962iT - 9.12e5T^{2} \) |
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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
−11.99099664498084366096494496948, −10.70839512519486331035953078984, −10.00742867766128240167464351816, −8.848087344515136778015542894947, −7.57310535014205494929287834914, −6.89490890536218954997138634845, −5.39785322726497494392861244111, −4.23616970281721007356707412749, −2.64511483351913043286358179974, −0.819866135306952500963460780247,
1.53802374098517992953125915134, 3.31134914687291845741414549016, 4.65384941255522121190455768147, 5.82532264956870446990620749060, 7.13261739251179517905256230814, 8.103018731224878711937240678754, 9.367714831589838942853812347706, 10.13344881585589638924205850822, 11.17379741080088000521611583393, 12.23389205751739045917647040635
