| L(s) = 1 | + 4.38i·3-s + 2.45·7-s − 10.2·9-s + 3.07i·11-s − 15.5i·13-s − 14.1·17-s − 35.3i·19-s + 10.7i·21-s + (−20.3 + 10.6i)23-s − 5.45i·27-s + 46.6·29-s − 12.3·31-s − 13.4·33-s + 65.0·37-s + 68.2·39-s + ⋯ |
| L(s) = 1 | + 1.46i·3-s + 0.350·7-s − 1.13·9-s + 0.279i·11-s − 1.19i·13-s − 0.833·17-s − 1.85i·19-s + 0.512i·21-s + (−0.886 + 0.463i)23-s − 0.202i·27-s + 1.60·29-s − 0.398·31-s − 0.409·33-s + 1.75·37-s + 1.75·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.998932327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.998932327\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + (20.3 - 10.6i)T \) |
| good | 3 | \( 1 - 4.38iT - 9T^{2} \) |
| 7 | \( 1 - 2.45T + 49T^{2} \) |
| 11 | \( 1 - 3.07iT - 121T^{2} \) |
| 13 | \( 1 + 15.5iT - 169T^{2} \) |
| 17 | \( 1 + 14.1T + 289T^{2} \) |
| 19 | \( 1 + 35.3iT - 361T^{2} \) |
| 29 | \( 1 - 46.6T + 841T^{2} \) |
| 31 | \( 1 + 12.3T + 961T^{2} \) |
| 37 | \( 1 - 65.0T + 1.36e3T^{2} \) |
| 41 | \( 1 - 36.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 68.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 72.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 79.0T + 2.80e3T^{2} \) |
| 59 | \( 1 - 15.6T + 3.48e3T^{2} \) |
| 61 | \( 1 - 0.819iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 86.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 20.1T + 5.04e3T^{2} \) |
| 73 | \( 1 - 34.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 113. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 88.9T + 6.88e3T^{2} \) |
| 89 | \( 1 - 50.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 29.7T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.271364779634350423130873891344, −8.259586318263333915454085064838, −7.59711340697445963772506184752, −6.46089176897530092357852577487, −5.61962521096426726420832094906, −4.56895301559424447171588411158, −4.50284740445350331816362928679, −3.20016087503322109612727328331, −2.45692662080505894083694328334, −0.69107183690017351563984426330,
0.78161938580074397997004747432, 1.81547083928246971753339897192, 2.42147021362345483011611068349, 3.84681490921534868482306420533, 4.67359747246800889536569703975, 6.04944128420381761813136374833, 6.29806762464784953703694820969, 7.17627384178119242884791774867, 7.998982958202949022640759764300, 8.357270394831885520764310599551
