L(s) = 1 | − 4·5-s − 8·7-s + 4·11-s − 6·13-s − 6·17-s + 4·19-s − 4·23-s + 11·25-s + 32·35-s − 8·37-s + 14·41-s − 16·43-s + 8·47-s + 34·49-s + 14·53-s − 16·55-s + 20·59-s + 24·65-s + 16·71-s − 32·77-s + 32·83-s + 24·85-s − 14·89-s + 48·91-s − 16·95-s + 12·103-s + 24·107-s + ⋯ |
L(s) = 1 | − 1.78·5-s − 3.02·7-s + 1.20·11-s − 1.66·13-s − 1.45·17-s + 0.917·19-s − 0.834·23-s + 11/5·25-s + 5.40·35-s − 1.31·37-s + 2.18·41-s − 2.43·43-s + 1.16·47-s + 34/7·49-s + 1.92·53-s − 2.15·55-s + 2.60·59-s + 2.97·65-s + 1.89·71-s − 3.64·77-s + 3.51·83-s + 2.60·85-s − 1.48·89-s + 5.03·91-s − 1.64·95-s + 1.18·103-s + 2.32·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6528257399\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6528257399\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
good | 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−9.105396186993257068585171424032, −8.926032643371771807066007268240, −8.495085274284798476093459736567, −8.050448779032864152808405759614, −7.33830184942043699395231898851, −7.20613114473201374929095604989, −6.85345074360539202771821235038, −6.78242932188826588277702428467, −6.06330833002352448655454551968, −5.95692402237381641419598458965, −4.98911975947275714817582857684, −4.88254915731970692211552497139, −4.01182135725353023197460861929, −3.90195398688430907611887589779, −3.47663098656798031285403818845, −3.24821677265466540154949836720, −2.35103085513481078882342672188, −2.33494398314945500993472276994, −0.68083492071063780409107803020, −0.44908064115229555464803894350,
0.44908064115229555464803894350, 0.68083492071063780409107803020, 2.33494398314945500993472276994, 2.35103085513481078882342672188, 3.24821677265466540154949836720, 3.47663098656798031285403818845, 3.90195398688430907611887589779, 4.01182135725353023197460861929, 4.88254915731970692211552497139, 4.98911975947275714817582857684, 5.95692402237381641419598458965, 6.06330833002352448655454551968, 6.78242932188826588277702428467, 6.85345074360539202771821235038, 7.20613114473201374929095604989, 7.33830184942043699395231898851, 8.050448779032864152808405759614, 8.495085274284798476093459736567, 8.926032643371771807066007268240, 9.105396186993257068585171424032