| L(s) = 1 | − 5·5-s − 32·7-s − 36·11-s + 10·13-s − 156·17-s + 280·19-s + 192·23-s − 6·29-s + 16·31-s + 160·35-s − 68·37-s + 390·41-s + 52·43-s − 408·47-s + 343·49-s − 228·53-s + 180·55-s − 516·59-s + 58·61-s − 50·65-s + 892·67-s − 240·71-s − 1.29e3·73-s + 1.15e3·77-s + 1.16e3·79-s + 732·83-s + 780·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.72·7-s − 0.986·11-s + 0.213·13-s − 2.22·17-s + 3.38·19-s + 1.74·23-s − 0.0384·29-s + 0.0926·31-s + 0.772·35-s − 0.302·37-s + 1.48·41-s + 0.184·43-s − 1.26·47-s + 49-s − 0.590·53-s + 0.441·55-s − 1.13·59-s + 0.121·61-s − 0.0954·65-s + 1.62·67-s − 0.401·71-s − 2.07·73-s + 1.70·77-s + 1.66·79-s + 0.968·83-s + 0.995·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4300909672\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4300909672\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 32 T + 681 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 36 T - 35 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 10 T - 2097 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 78 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 140 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 192 T + 24697 T^{2} - 192 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T - 24353 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 16 T - 29535 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 34 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 390 T + 83179 T^{2} - 390 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 52 T - 76803 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 408 T + 62641 T^{2} + 408 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 114 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 516 T + 60877 T^{2} + 516 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 58 T - 223617 T^{2} - 58 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 892 T + 494901 T^{2} - 892 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 120 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 646 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 1168 T + 871185 T^{2} - 1168 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 732 T - 35963 T^{2} - 732 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1590 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 p T - 93 p^{2} T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
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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
−9.406387598840837422606943302934, −8.905214130167463235394567076259, −8.572304199300811164115246499573, −7.87076860642671868093241691932, −7.63425758583522817761418093833, −7.11313868457685131755454460130, −6.98515823862389530334674548674, −6.38292251657889686978263648065, −6.16464000749535693100044808661, −5.35526127206624336186457874705, −5.27907501843995660221541260089, −4.69223559190082001659282704677, −4.22749967939101336837958978426, −3.41899285004078726050448338984, −3.35255608716839868316821000213, −2.67356631562821885401949476438, −2.61576544502341411135940324774, −1.45812535257330963835367908214, −0.916182371466648722003163065475, −0.17421488488855393119858874445,
0.17421488488855393119858874445, 0.916182371466648722003163065475, 1.45812535257330963835367908214, 2.61576544502341411135940324774, 2.67356631562821885401949476438, 3.35255608716839868316821000213, 3.41899285004078726050448338984, 4.22749967939101336837958978426, 4.69223559190082001659282704677, 5.27907501843995660221541260089, 5.35526127206624336186457874705, 6.16464000749535693100044808661, 6.38292251657889686978263648065, 6.98515823862389530334674548674, 7.11313868457685131755454460130, 7.63425758583522817761418093833, 7.87076860642671868093241691932, 8.572304199300811164115246499573, 8.905214130167463235394567076259, 9.406387598840837422606943302934
