| L(s) = 1 | + 3·3-s − 2·5-s − 4·7-s + 6·9-s − 6·15-s + 6·17-s − 12·21-s + 3·25-s + 9·27-s + 8·35-s − 16·37-s + 12·41-s − 20·43-s − 12·45-s − 6·47-s + 9·49-s + 18·51-s − 12·59-s − 24·63-s − 4·67-s + 9·75-s + 26·79-s + 9·81-s + 24·83-s − 12·85-s + 36·101-s + 24·105-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.894·5-s − 1.51·7-s + 2·9-s − 1.54·15-s + 1.45·17-s − 2.61·21-s + 3/5·25-s + 1.73·27-s + 1.35·35-s − 2.63·37-s + 1.87·41-s − 3.04·43-s − 1.78·45-s − 0.875·47-s + 9/7·49-s + 2.52·51-s − 1.56·59-s − 3.02·63-s − 0.488·67-s + 1.03·75-s + 2.92·79-s + 81-s + 2.63·83-s − 1.30·85-s + 3.58·101-s + 2.34·105-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.499663072\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.499663072\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 191 T^{2} + p^{2} T^{4} \) |
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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
−9.733749040158929515548400794686, −9.050455057028632390964115449926, −8.851558402607513733528858424549, −8.356976894471798705370994653717, −7.911767255622625375704060215739, −7.68414124888335670709591020634, −7.40190163913803069998591671638, −6.67623566099226817054563724135, −6.63585771449316877949278701974, −6.12237931287808580117975900809, −5.29591712777220062768202091441, −5.01702267698658122950743633720, −4.41701088323047055941457096225, −3.66216343744557861561566676306, −3.54415772029358230774224103448, −3.26450710803956470259550416904, −2.88663872646298472636726172871, −2.08476272030609904313013749633, −1.55202335156257597807165764069, −0.53656611710741931084683969148,
0.53656611710741931084683969148, 1.55202335156257597807165764069, 2.08476272030609904313013749633, 2.88663872646298472636726172871, 3.26450710803956470259550416904, 3.54415772029358230774224103448, 3.66216343744557861561566676306, 4.41701088323047055941457096225, 5.01702267698658122950743633720, 5.29591712777220062768202091441, 6.12237931287808580117975900809, 6.63585771449316877949278701974, 6.67623566099226817054563724135, 7.40190163913803069998591671638, 7.68414124888335670709591020634, 7.911767255622625375704060215739, 8.356976894471798705370994653717, 8.851558402607513733528858424549, 9.050455057028632390964115449926, 9.733749040158929515548400794686
