| L(s) = 1 | − 3-s + 7-s + 9-s − 11-s + 4·13-s − 17-s − 21-s − 9·23-s − 5·25-s − 27-s + 3·29-s + 10·31-s + 33-s − 2·37-s − 4·39-s + 6·41-s − 9·43-s + 47-s − 6·49-s + 51-s − 8·53-s − 8·59-s − 2·61-s + 63-s + 8·67-s + 9·69-s − 3·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s − 0.242·17-s − 0.218·21-s − 1.87·23-s − 25-s − 0.192·27-s + 0.557·29-s + 1.79·31-s + 0.174·33-s − 0.328·37-s − 0.640·39-s + 0.937·41-s − 1.37·43-s + 0.145·47-s − 6/7·49-s + 0.140·51-s − 1.09·53-s − 1.04·59-s − 0.256·61-s + 0.125·63-s + 0.977·67-s + 1.08·69-s − 0.356·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−14.00771566904902, −13.46140911489251, −13.28031230027042, −12.38516644163320, −12.01887572615717, −11.70251843529165, −11.08849511768340, −10.59580170690134, −10.27466776392508, −9.557981862815372, −9.261836210542564, −8.256415910886108, −8.051471455428952, −7.791159095913497, −6.710305532002442, −6.301722476714941, −6.118025312873235, −5.243965068198040, −4.864341487019242, −4.145050834203537, −3.754065712087681, −2.986994042117042, −2.151279924778351, −1.633720932161928, −0.8441875386712860, 0,
0.8441875386712860, 1.633720932161928, 2.151279924778351, 2.986994042117042, 3.754065712087681, 4.145050834203537, 4.864341487019242, 5.243965068198040, 6.118025312873235, 6.301722476714941, 6.710305532002442, 7.791159095913497, 8.051471455428952, 8.256415910886108, 9.261836210542564, 9.557981862815372, 10.27466776392508, 10.59580170690134, 11.08849511768340, 11.70251843529165, 12.01887572615717, 12.38516644163320, 13.28031230027042, 13.46140911489251, 14.00771566904902
