| L(s) = 1 | + 8·2-s + 48·4-s − 14·5-s + 256·8-s − 112·10-s + 238·13-s + 1.28e3·16-s + 322·17-s − 672·20-s + 625·25-s + 1.90e3·26-s + 82·29-s + 6.14e3·32-s + 2.57e3·34-s − 2.16e3·37-s − 3.58e3·40-s + 6.07e3·41-s + 4.80e3·49-s + 5.00e3·50-s + 1.14e4·52-s − 4.96e3·53-s + 656·58-s + 6.95e3·61-s + 2.86e4·64-s − 3.33e3·65-s + 1.54e4·68-s − 1.44e3·73-s + ⋯ |
| L(s) = 1 | + 2·2-s + 3·4-s − 0.559·5-s + 4·8-s − 1.11·10-s + 1.40·13-s + 5·16-s + 1.11·17-s − 1.67·20-s + 25-s + 2.81·26-s + 0.0975·29-s + 6·32-s + 2.22·34-s − 1.57·37-s − 2.23·40-s + 3.61·41-s + 2·49-s + 2·50-s + 4.22·52-s − 1.76·53-s + 0.195·58-s + 1.86·61-s + 7·64-s − 0.788·65-s + 3.34·68-s − 0.270·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(15.82356275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.82356275\) |
| \(L(3)\) |
|
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 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 14 T - 429 T^{2} + 14 p^{4} T^{3} + p^{8} T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 238 T + 28083 T^{2} - 238 p^{4} T^{3} + p^{8} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 322 T + 20163 T^{2} - 322 p^{4} T^{3} + p^{8} T^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 82 T - 700557 T^{2} - 82 p^{4} T^{3} + p^{8} T^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2162 T + 2800083 T^{2} + 2162 p^{4} T^{3} + p^{8} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3038 T + p^{4} T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2482 T + p^{4} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 6958 T + 34567923 T^{2} - 6958 p^{4} T^{3} + p^{8} T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 1442 T - 26318877 T^{2} + 1442 p^{4} T^{3} + p^{8} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 9758 T + 32476323 T^{2} + 9758 p^{4} T^{3} + p^{8} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1918 T + p^{4} T^{2} )^{2} \) |
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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
−11.15552742781605896148988918639, −11.01544977426053158600641150875, −10.68318345298913576105065499511, −10.10532794904752146487640096441, −9.422590321485933758714601150406, −8.727433759371784516594378279083, −8.020041542912668898446847782487, −7.84876023387037231282894836543, −7.01462638174838718609236820568, −6.88259237217843305262971185029, −6.09589137344184972083417641024, −5.65930335961725304584687750589, −5.34957254311324661746055510835, −4.54777949069657368984599226850, −3.95267475559926789066948632716, −3.75757995424259221035458687473, −2.95377941956183330614741943774, −2.52426779051190979916421180380, −1.42079023001351098158559261117, −0.947670359894114978696696432705,
0.947670359894114978696696432705, 1.42079023001351098158559261117, 2.52426779051190979916421180380, 2.95377941956183330614741943774, 3.75757995424259221035458687473, 3.95267475559926789066948632716, 4.54777949069657368984599226850, 5.34957254311324661746055510835, 5.65930335961725304584687750589, 6.09589137344184972083417641024, 6.88259237217843305262971185029, 7.01462638174838718609236820568, 7.84876023387037231282894836543, 8.020041542912668898446847782487, 8.727433759371784516594378279083, 9.422590321485933758714601150406, 10.10532794904752146487640096441, 10.68318345298913576105065499511, 11.01544977426053158600641150875, 11.15552742781605896148988918639
