| L(s) = 1 | − 3.87·3-s + 7.74·7-s + 6.00·9-s + 19.3i·11-s + 20i·13-s + 15i·17-s − 19.3i·19-s − 30.0·21-s + 7.74·23-s + 11.6·27-s − 48·29-s + 38.7i·31-s − 75i·33-s − 10i·37-s − 77.4i·39-s + ⋯ |
| L(s) = 1 | − 1.29·3-s + 1.10·7-s + 0.666·9-s + 1.76i·11-s + 1.53i·13-s + 0.882i·17-s − 1.01i·19-s − 1.42·21-s + 0.336·23-s + 0.430·27-s − 1.65·29-s + 1.24i·31-s − 2.27i·33-s − 0.270i·37-s − 1.98i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8432000960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8432000960\) |
| \(L(2)\) |
|
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 | \( 1 + 3.87T + 9T^{2} \) |
| 7 | \( 1 - 7.74T + 49T^{2} \) |
| 11 | \( 1 - 19.3iT - 121T^{2} \) |
| 13 | \( 1 - 20iT - 169T^{2} \) |
| 17 | \( 1 - 15iT - 289T^{2} \) |
| 19 | \( 1 + 19.3iT - 361T^{2} \) |
| 23 | \( 1 - 7.74T + 529T^{2} \) |
| 29 | \( 1 + 48T + 841T^{2} \) |
| 31 | \( 1 - 38.7iT - 961T^{2} \) |
| 37 | \( 1 + 10iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 33T + 1.68e3T^{2} \) |
| 43 | \( 1 - 61.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 15.4T + 2.20e3T^{2} \) |
| 53 | \( 1 - 30iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 77.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 38T + 3.72e3T^{2} \) |
| 67 | \( 1 - 58.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 77.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 5iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 38.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 11.6T + 6.88e3T^{2} \) |
| 89 | \( 1 + 87T + 7.92e3T^{2} \) |
| 97 | \( 1 - 110iT - 9.40e3T^{2} \) |
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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
−9.518802953541676234390940888516, −8.955258390354015144205395692439, −7.73297116639899257737151900096, −7.03939806142326037534234941219, −6.39455514362583267214071117798, −5.29427686925085155228779869323, −4.71596805036242745330958941323, −4.09123257370951065900690469635, −2.17432202706091098180604775635, −1.39979129139849470909811851194,
0.31689888532818608774303986169, 1.13465012662652418593623766541, 2.76255100911814259546111199604, 3.88689899157935449107905007840, 5.05095204708422267275430325455, 5.75102941605345725512902450099, 5.94363066743398669093088876274, 7.39276006606033371735821422056, 7.993441183151880672368709656062, 8.758905641376691795345719770751
