| L(s) = 1 | − 2.44i·5-s + 3.46·7-s + 8.48i·11-s + 10.3·13-s − 12.7i·17-s − 4·19-s + 34.2i·23-s + 19·25-s + 46.5i·29-s − 38.1·31-s − 8.48i·35-s − 27.7·37-s + 29.6i·41-s + 56·43-s − 63.6i·47-s + ⋯ |
| L(s) = 1 | − 0.489i·5-s + 0.494·7-s + 0.771i·11-s + 0.799·13-s − 0.748i·17-s − 0.210·19-s + 1.49i·23-s + 0.760·25-s + 1.60i·29-s − 1.22·31-s − 0.242i·35-s − 0.748·37-s + 0.724i·41-s + 1.30·43-s − 1.35i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.992036352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.992036352\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2.44iT - 25T^{2} \) |
| 7 | \( 1 - 3.46T + 49T^{2} \) |
| 11 | \( 1 - 8.48iT - 121T^{2} \) |
| 13 | \( 1 - 10.3T + 169T^{2} \) |
| 17 | \( 1 + 12.7iT - 289T^{2} \) |
| 19 | \( 1 + 4T + 361T^{2} \) |
| 23 | \( 1 - 34.2iT - 529T^{2} \) |
| 29 | \( 1 - 46.5iT - 841T^{2} \) |
| 31 | \( 1 + 38.1T + 961T^{2} \) |
| 37 | \( 1 + 27.7T + 1.36e3T^{2} \) |
| 41 | \( 1 - 29.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 56T + 1.84e3T^{2} \) |
| 47 | \( 1 + 63.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 71.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 101. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 69.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 28T + 4.48e3T^{2} \) |
| 71 | \( 1 - 53.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 20T + 5.32e3T^{2} \) |
| 79 | \( 1 + 72.7T + 6.24e3T^{2} \) |
| 83 | \( 1 - 76.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 55.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 8T + 9.40e3T^{2} \) |
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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
−8.921856949824969386996707984963, −8.290681879999281174356853430946, −7.29292445784367895110686662596, −6.85226065022996953527296791254, −5.50748534319640712521198537438, −5.13469269158588508528941950604, −4.12414046800898171610769724313, −3.24398810914439047941812494026, −1.93017565991734420888557309383, −1.08994152759511784126989150370,
0.53661356136191090791361848151, 1.82087483102853739140150284997, 2.86606095734531973228216779945, 3.82362464643701532026650416746, 4.60185322990476760025963789374, 5.78569694249285007533021258339, 6.25213029851470840540613577004, 7.15042549603559816891844818004, 8.102123947090219488782944693058, 8.567350254586394909922942503098
