L(s) = 1 | + 3-s − 2·5-s + 2·7-s + 3·9-s − 7·11-s + 3·13-s − 2·15-s + 5·17-s − 2·19-s + 2·21-s − 2·23-s + 3·25-s + 8·27-s − 3·29-s + 16·31-s − 7·33-s − 4·35-s − 4·37-s + 3·39-s + 2·41-s + 6·43-s − 6·45-s − 3·47-s + 3·49-s + 5·51-s + 10·53-s + 14·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s + 0.755·7-s + 9-s − 2.11·11-s + 0.832·13-s − 0.516·15-s + 1.21·17-s − 0.458·19-s + 0.436·21-s − 0.417·23-s + 3/5·25-s + 1.53·27-s − 0.557·29-s + 2.87·31-s − 1.21·33-s − 0.676·35-s − 0.657·37-s + 0.480·39-s + 0.312·41-s + 0.914·43-s − 0.894·45-s − 0.437·47-s + 3/7·49-s + 0.700·51-s + 1.37·53-s + 1.88·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.059216868\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.059216868\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 5 T + 32 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 50 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 62 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 10 T + 98 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 13 T + 126 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 226 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 9 T + 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−10.84284078540098341626963195467, −10.35786564446938164222143662063, −10.27327239549735999303343305855, −9.944640394319430645146724392394, −8.972040494894194956693592046757, −8.710640075836920442172871717424, −8.165936620840861744646453861047, −7.920118614036728683234225514056, −7.62317669511692236099655583520, −7.18868884605678539838150836467, −6.50284999339652383658178795669, −5.90263395361255143378942778550, −5.35530612303591521851006727523, −4.74843801267861323629419970636, −4.39541525840451071822739290017, −3.87198230778114132426753634651, −2.87980350609293473307188815671, −2.87852072970228192430237051573, −1.76637398585676073504220174683, −0.846791506230275215589283923994,
0.846791506230275215589283923994, 1.76637398585676073504220174683, 2.87852072970228192430237051573, 2.87980350609293473307188815671, 3.87198230778114132426753634651, 4.39541525840451071822739290017, 4.74843801267861323629419970636, 5.35530612303591521851006727523, 5.90263395361255143378942778550, 6.50284999339652383658178795669, 7.18868884605678539838150836467, 7.62317669511692236099655583520, 7.920118614036728683234225514056, 8.165936620840861744646453861047, 8.710640075836920442172871717424, 8.972040494894194956693592046757, 9.944640394319430645146724392394, 10.27327239549735999303343305855, 10.35786564446938164222143662063, 10.84284078540098341626963195467