| L(s) = 1 | − 3·3-s + 4·5-s + 20·11-s + 8·13-s − 12·15-s + 24·17-s + 44·19-s − 72·23-s + 125·25-s + 27·27-s − 76·29-s + 184·31-s − 60·33-s + 30·37-s − 24·39-s + 432·41-s − 328·43-s + 520·47-s − 72·51-s + 146·53-s + 80·55-s − 132·57-s + 460·59-s + 628·61-s + 32·65-s − 556·67-s + 216·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.357·5-s + 0.548·11-s + 0.170·13-s − 0.206·15-s + 0.342·17-s + 0.531·19-s − 0.652·23-s + 25-s + 0.192·27-s − 0.486·29-s + 1.06·31-s − 0.316·33-s + 0.133·37-s − 0.0985·39-s + 1.64·41-s − 1.16·43-s + 1.61·47-s − 0.197·51-s + 0.378·53-s + 0.196·55-s − 0.306·57-s + 1.01·59-s + 1.31·61-s + 0.0610·65-s − 1.01·67-s + 0.376·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.272011201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.272011201\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T - 109 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 20 T - 931 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 24 T - 4337 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 44 T - 4923 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 72 T - 6983 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 38 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 184 T + 4065 T^{2} - 184 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 30 T - 49753 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 216 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 164 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 520 T + 166577 T^{2} - 520 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 146 T - 127561 T^{2} - 146 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 460 T + 6221 T^{2} - 460 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 628 T + 167403 T^{2} - 628 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 556 T + 8373 T^{2} + 556 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 592 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 1024 T + 659559 T^{2} - 1024 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 104 T - 482223 T^{2} - 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 324 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 896 T + 97847 T^{2} - 896 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 920 T + p^{3} T^{2} )^{2} \) |
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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
−10.29756792069930784077773590280, −10.28501472986141067964717362583, −9.579509974861709741916292958655, −9.366696393513900621970042912594, −8.633058739216027614299548621656, −8.524805623920164019941017999718, −7.68318560042889639354538152357, −7.48580460847140619301948258256, −6.73493527671600003524256422709, −6.49030756715918419417843177882, −5.81540049302534786667771493002, −5.67987843979867536561124685037, −4.82818056158691188736877355282, −4.66798234076906158018235167844, −3.65654234049244366947305095427, −3.53110357545327346674506742778, −2.47159855915541263947291499495, −2.08201840542395782984467963852, −0.927249346413804176962241745199, −0.75041644127924798445666235203,
0.75041644127924798445666235203, 0.927249346413804176962241745199, 2.08201840542395782984467963852, 2.47159855915541263947291499495, 3.53110357545327346674506742778, 3.65654234049244366947305095427, 4.66798234076906158018235167844, 4.82818056158691188736877355282, 5.67987843979867536561124685037, 5.81540049302534786667771493002, 6.49030756715918419417843177882, 6.73493527671600003524256422709, 7.48580460847140619301948258256, 7.68318560042889639354538152357, 8.524805623920164019941017999718, 8.633058739216027614299548621656, 9.366696393513900621970042912594, 9.579509974861709741916292958655, 10.28501472986141067964717362583, 10.29756792069930784077773590280
