| L(s) = 1 | + 4·2-s + 2·3-s + 12·4-s − 10·5-s + 8·6-s − 16·7-s + 32·8-s − 45·9-s − 40·10-s + 90·11-s + 24·12-s − 50·13-s − 64·14-s − 20·15-s + 80·16-s + 44·17-s − 180·18-s − 108·19-s − 120·20-s − 32·21-s + 360·22-s − 28·23-s + 64·24-s + 41·25-s − 200·26-s − 134·27-s − 192·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.384·3-s + 3/2·4-s − 0.894·5-s + 0.544·6-s − 0.863·7-s + 1.41·8-s − 5/3·9-s − 1.26·10-s + 2.46·11-s + 0.577·12-s − 1.06·13-s − 1.22·14-s − 0.344·15-s + 5/4·16-s + 0.627·17-s − 2.35·18-s − 1.30·19-s − 1.34·20-s − 0.332·21-s + 3.48·22-s − 0.253·23-s + 0.544·24-s + 0.327·25-s − 1.50·26-s − 0.955·27-s − 1.29·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2829124 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2829124 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 29 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 49 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 p T + 59 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 16 T + 366 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 90 T + 4633 T^{2} - 90 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 50 T + 4635 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 44 T + 8774 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 108 T + 16538 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 28 T - 11974 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 66 T - 10615 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 40 T + 101610 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 304 T + 122546 T^{2} + 304 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 130 T + 146385 T^{2} - 130 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 514 T + 214889 T^{2} - 514 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 958 T + 510971 T^{2} + 958 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 180 T + 43858 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1028 T + 717294 T^{2} + 1028 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 912 T + 807062 T^{2} + 912 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 796 T + 867290 T^{2} - 796 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 856 T + 775362 T^{2} - 856 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 318 T + 229433 T^{2} - 318 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1828 T + 1970306 T^{2} + 1828 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 944 T + 1601618 T^{2} - 944 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 368 T + 1799202 T^{2} + 368 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.854666529908259088877266004036, −8.357459420570986983430868565558, −7.76988079725919153404944185418, −7.70146318664463854031268804806, −6.97720122297345541946155170602, −6.67354799083824333378933648998, −6.19204900942041565699404116757, −6.14011352050397520907883956009, −5.58526447152926549405345764488, −4.89296138084937115813747441373, −4.63409825890299690505222187276, −4.03148256964322024240479455027, −3.59991549050402697535219287771, −3.51591828231021456502152134984, −2.85333699546830435676548938648, −2.53141195951748220669885334099, −1.77854794488332429778878024590, −1.21611232630329423102493202622, 0, 0,
1.21611232630329423102493202622, 1.77854794488332429778878024590, 2.53141195951748220669885334099, 2.85333699546830435676548938648, 3.51591828231021456502152134984, 3.59991549050402697535219287771, 4.03148256964322024240479455027, 4.63409825890299690505222187276, 4.89296138084937115813747441373, 5.58526447152926549405345764488, 6.14011352050397520907883956009, 6.19204900942041565699404116757, 6.67354799083824333378933648998, 6.97720122297345541946155170602, 7.70146318664463854031268804806, 7.76988079725919153404944185418, 8.357459420570986983430868565558, 8.854666529908259088877266004036
