| L(s) = 1 | − 3·3-s + 11·5-s − 7·7-s + 39·11-s − 64·13-s − 33·15-s − 12·17-s − 88·19-s + 21·21-s − 92·23-s + 125·25-s + 27·27-s + 510·29-s − 35·31-s − 117·33-s − 77·35-s + 4·37-s + 192·39-s + 32·41-s + 660·43-s − 298·47-s − 294·49-s + 36·51-s + 717·53-s + 429·55-s + 264·57-s − 217·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.983·5-s − 0.377·7-s + 1.06·11-s − 1.36·13-s − 0.568·15-s − 0.171·17-s − 1.06·19-s + 0.218·21-s − 0.834·23-s + 25-s + 0.192·27-s + 3.26·29-s − 0.202·31-s − 0.617·33-s − 0.371·35-s + 0.0177·37-s + 0.788·39-s + 0.121·41-s + 2.34·43-s − 0.924·47-s − 6/7·49-s + 0.0988·51-s + 1.85·53-s + 1.05·55-s + 0.613·57-s − 0.478·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.015909382\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.015909382\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | $C_2$ | \( 1 + p T + p^{3} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 11 T - 4 T^{2} - 11 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 39 T + 190 T^{2} - 39 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 32 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 12 T - 4769 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 88 T + 885 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 p T - 7 p^{2} T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 255 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 35 T - 28566 T^{2} + 35 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 4 T - 50637 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 16 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 330 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 298 T - 15019 T^{2} + 298 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 717 T + 365212 T^{2} - 717 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 217 T - 158290 T^{2} + 217 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 386 T - 77985 T^{2} + 386 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 906 T + 520073 T^{2} - 906 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 34 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 838 T + 313227 T^{2} - 838 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 1325 T + 1262586 T^{2} - 1325 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 1163 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 54 T - 702053 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p^{3} T^{2} )^{2} \) |
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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
−11.56641506859709831212182368922, −10.71424970971433609709336347901, −10.33467652471693386298485957650, −10.14308912984771705767992578502, −9.378067486121033348275837944038, −9.328713661250548137451273626901, −8.542782495401402277838636117429, −8.197116351964591429807290406992, −7.42619679551231689507823702189, −6.77836115154805281735638699688, −6.37858696401191257840919672184, −6.27038746254520979745125509885, −5.43944243313879314411465744454, −4.90976003559617394189440768795, −4.40444983285209935531387623713, −3.79353454612816291780048322034, −2.59342736091541789983787832525, −2.49148217879357486636863654672, −1.37345297973467465439409282958, −0.55067484885553174790036545406,
0.55067484885553174790036545406, 1.37345297973467465439409282958, 2.49148217879357486636863654672, 2.59342736091541789983787832525, 3.79353454612816291780048322034, 4.40444983285209935531387623713, 4.90976003559617394189440768795, 5.43944243313879314411465744454, 6.27038746254520979745125509885, 6.37858696401191257840919672184, 6.77836115154805281735638699688, 7.42619679551231689507823702189, 8.197116351964591429807290406992, 8.542782495401402277838636117429, 9.328713661250548137451273626901, 9.378067486121033348275837944038, 10.14308912984771705767992578502, 10.33467652471693386298485957650, 10.71424970971433609709336347901, 11.56641506859709831212182368922
