| L(s) = 1 | + 3·2-s + 8·4-s − 5·5-s − 28·7-s + 45·8-s − 15·10-s − 45·11-s − 62·13-s − 84·14-s + 135·16-s + 96·17-s − 149·19-s − 40·20-s − 135·22-s − 141·23-s − 186·26-s − 224·28-s − 96·29-s + 178·31-s + 360·32-s + 288·34-s + 140·35-s − 371·37-s − 447·38-s − 225·40-s − 450·41-s + 688·43-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 4-s − 0.447·5-s − 1.51·7-s + 1.98·8-s − 0.474·10-s − 1.23·11-s − 1.32·13-s − 1.60·14-s + 2.10·16-s + 1.36·17-s − 1.79·19-s − 0.447·20-s − 1.30·22-s − 1.27·23-s − 1.40·26-s − 1.51·28-s − 0.614·29-s + 1.03·31-s + 1.98·32-s + 1.45·34-s + 0.676·35-s − 1.64·37-s − 1.90·38-s − 0.889·40-s − 1.71·41-s + 2.43·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.849350716\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.849350716\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 p T + p^{3} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 45 T + 694 T^{2} + 45 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 31 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 96 T + 4303 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 149 T + 15342 T^{2} + 149 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 141 T + 7714 T^{2} + 141 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 48 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 178 T + 1893 T^{2} - 178 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 371 T + 86988 T^{2} + 371 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 225 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 p T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 375 T + 36802 T^{2} - 375 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 663 T + 290692 T^{2} + 663 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 60 T - 201779 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 392 T - 73317 T^{2} + 392 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 280 T - 222363 T^{2} - 280 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 258 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 578 T - 54933 T^{2} + 578 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 152 T - 469935 T^{2} + 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 432 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 234 T - 650213 T^{2} + 234 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 1352 T + p^{3} T^{2} )^{2} \) |
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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
−11.67000932331609470538717306045, −10.90314099728175477681076623042, −10.40855174708192625495862960806, −10.13735283189505825680527148080, −10.08876664059676676704500493823, −9.168683459537615058369005143479, −8.509612065925353101001937136763, −7.73505997731891781834010482215, −7.60875077996572215531549785611, −7.22648286409740597248600214279, −6.33422132796655307636554537345, −6.17478016213491399478531554983, −5.37204861817247955546814817876, −4.91273954369246719716657826193, −4.31234709070885863053314178413, −3.75241315414574802268177346336, −3.14416727232988715152437336857, −2.47211312391143959295759430730, −1.84944145226318900062138536997, −0.38976606010377497108393522248,
0.38976606010377497108393522248, 1.84944145226318900062138536997, 2.47211312391143959295759430730, 3.14416727232988715152437336857, 3.75241315414574802268177346336, 4.31234709070885863053314178413, 4.91273954369246719716657826193, 5.37204861817247955546814817876, 6.17478016213491399478531554983, 6.33422132796655307636554537345, 7.22648286409740597248600214279, 7.60875077996572215531549785611, 7.73505997731891781834010482215, 8.509612065925353101001937136763, 9.168683459537615058369005143479, 10.08876664059676676704500493823, 10.13735283189505825680527148080, 10.40855174708192625495862960806, 10.90314099728175477681076623042, 11.67000932331609470538717306045
