| L(s) = 1 | − 2·2-s − 5·5-s − 14·7-s + 8·8-s + 10·10-s + 6·11-s − 68·13-s + 28·14-s − 16·16-s − 156·17-s + 88·19-s − 12·22-s + 120·23-s + 136·26-s + 126·29-s + 244·31-s + 312·34-s + 70·35-s − 608·37-s − 176·38-s − 40·40-s − 480·41-s − 104·43-s − 240·46-s + 600·47-s + 343·49-s + 516·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.447·5-s − 0.755·7-s + 0.353·8-s + 0.316·10-s + 0.164·11-s − 1.45·13-s + 0.534·14-s − 1/4·16-s − 2.22·17-s + 1.06·19-s − 0.116·22-s + 1.08·23-s + 1.02·26-s + 0.806·29-s + 1.41·31-s + 1.57·34-s + 0.338·35-s − 2.70·37-s − 0.751·38-s − 0.158·40-s − 1.82·41-s − 0.368·43-s − 0.769·46-s + 1.86·47-s + 49-s + 1.33·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.022292925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.022292925\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 p T - 3 p^{2} T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T - 1295 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 68 T + 2427 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 78 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 44 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 120 T + 2233 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 126 T - 8513 T^{2} - 126 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 244 T + 29745 T^{2} - 244 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 304 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 480 T + 161479 T^{2} + 480 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 104 T - 68691 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 600 T + 256177 T^{2} - 600 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 258 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 534 T + 79777 T^{2} - 534 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 362 T - 95937 T^{2} + 362 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 p T - 51 p^{2} T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 972 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 470 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 1244 T + 1054497 T^{2} + 1244 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 396 T - 414971 T^{2} - 396 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 972 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 46 T - 910557 T^{2} - 46 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.953484201120309725964394893902, −9.723226220786859470012486102080, −9.035196337251817847602833911429, −8.934941243252355244726807936089, −8.515850179315590759417505213659, −8.048376392844444442410170537612, −7.44341749514887886407294045500, −6.96378722362183589642910148184, −6.72567292619690724621766534315, −6.60112852050669371496419690328, −5.52153334075868838309731534712, −5.06998141274431255076505812897, −4.84456105884457559984360976997, −4.12057393426852355791066510729, −3.64036039217303153172313992457, −2.99504874152822162071524140063, −2.40494693836797730481215758588, −1.93059209658484318798592657299, −0.78096319082402285598252826912, −0.43419380193689987061901834077,
0.43419380193689987061901834077, 0.78096319082402285598252826912, 1.93059209658484318798592657299, 2.40494693836797730481215758588, 2.99504874152822162071524140063, 3.64036039217303153172313992457, 4.12057393426852355791066510729, 4.84456105884457559984360976997, 5.06998141274431255076505812897, 5.52153334075868838309731534712, 6.60112852050669371496419690328, 6.72567292619690724621766534315, 6.96378722362183589642910148184, 7.44341749514887886407294045500, 8.048376392844444442410170537612, 8.515850179315590759417505213659, 8.934941243252355244726807936089, 9.035196337251817847602833911429, 9.723226220786859470012486102080, 9.953484201120309725964394893902
