| L(s) = 1 | + 12.7·5-s − 14.0·7-s − 42.5·11-s − 72.4·13-s − 59.6·17-s + 105.·19-s − 0.225·23-s + 37.3·25-s − 225.·29-s + 201.·31-s − 179.·35-s − 152.·37-s − 489.·41-s − 7.58·43-s + 373.·47-s − 145.·49-s − 43.6·53-s − 542.·55-s − 671.·59-s + 74.0·61-s − 923.·65-s + 420.·67-s − 730.·71-s + 473.·73-s + 597.·77-s − 529.·79-s + 26.1·83-s + ⋯ |
| L(s) = 1 | + 1.13·5-s − 0.758·7-s − 1.16·11-s − 1.54·13-s − 0.850·17-s + 1.27·19-s − 0.00204·23-s + 0.298·25-s − 1.44·29-s + 1.16·31-s − 0.864·35-s − 0.679·37-s − 1.86·41-s − 0.0269·43-s + 1.15·47-s − 0.424·49-s − 0.113·53-s − 1.32·55-s − 1.48·59-s + 0.155·61-s − 1.76·65-s + 0.767·67-s − 1.22·71-s + 0.759·73-s + 0.884·77-s − 0.754·79-s + 0.0345·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 12.7T + 125T^{2} \) |
| 7 | \( 1 + 14.0T + 343T^{2} \) |
| 11 | \( 1 + 42.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 72.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 59.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 105.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 0.225T + 1.21e4T^{2} \) |
| 29 | \( 1 + 225.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 201.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 152.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 489.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 7.58T + 7.95e4T^{2} \) |
| 47 | \( 1 - 373.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 43.6T + 1.48e5T^{2} \) |
| 59 | \( 1 + 671.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 74.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 420.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 730.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 473.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 529.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 26.1T + 5.71e5T^{2} \) |
| 89 | \( 1 - 415.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 927.T + 9.12e5T^{2} \) |
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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
−10.38900532666365602113775101160, −9.855030185168319982453673901183, −9.127346057678020024502385748460, −7.72231481723388174492355720391, −6.81117418023576809461801940969, −5.66535193921582085225455563071, −4.89001282112252787716928740941, −3.06813069151916590019877419157, −2.06994792899353805548013460186, 0,
2.06994792899353805548013460186, 3.06813069151916590019877419157, 4.89001282112252787716928740941, 5.66535193921582085225455563071, 6.81117418023576809461801940969, 7.72231481723388174492355720391, 9.127346057678020024502385748460, 9.855030185168319982453673901183, 10.38900532666365602113775101160
