| L(s) = 1 | − 2·3-s + 2·5-s + 3·9-s − 2·11-s + 4·13-s − 4·15-s + 4·17-s + 3·25-s − 4·27-s + 4·29-s + 4·33-s + 12·37-s − 8·39-s − 4·41-s + 8·43-s + 6·45-s − 6·49-s − 8·51-s + 12·53-s − 4·55-s + 8·59-s + 12·61-s + 8·65-s + 8·67-s − 16·71-s + 12·73-s − 6·75-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 9-s − 0.603·11-s + 1.10·13-s − 1.03·15-s + 0.970·17-s + 3/5·25-s − 0.769·27-s + 0.742·29-s + 0.696·33-s + 1.97·37-s − 1.28·39-s − 0.624·41-s + 1.21·43-s + 0.894·45-s − 6/7·49-s − 1.12·51-s + 1.64·53-s − 0.539·55-s + 1.04·59-s + 1.53·61-s + 0.992·65-s + 0.977·67-s − 1.89·71-s + 1.40·73-s − 0.692·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.235043492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.235043492\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 118 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 222 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 20 T + 262 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
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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
−9.887388423961949444985083610234, −9.678230721220770104624764288567, −8.912939053555486145167316347377, −8.839570356463871244375101962759, −8.116553803398584272437994383577, −7.79727508449018962378187484249, −7.39512485425364474919536425619, −6.77691030309643855630134962909, −6.41160120121493841127413537804, −6.09688042198451277784963774623, −5.63248807025867468103369325878, −5.43681065177821883829560299210, −4.77541542109312784994948306351, −4.55439792171133255601234232385, −3.58890051212256976367339351706, −3.53652557798013697486396483554, −2.41880531374975573859986365190, −2.21312601188852818935922061661, −1.06263566468414675787265062658, −0.863273212180591144309694211998,
0.863273212180591144309694211998, 1.06263566468414675787265062658, 2.21312601188852818935922061661, 2.41880531374975573859986365190, 3.53652557798013697486396483554, 3.58890051212256976367339351706, 4.55439792171133255601234232385, 4.77541542109312784994948306351, 5.43681065177821883829560299210, 5.63248807025867468103369325878, 6.09688042198451277784963774623, 6.41160120121493841127413537804, 6.77691030309643855630134962909, 7.39512485425364474919536425619, 7.79727508449018962378187484249, 8.116553803398584272437994383577, 8.839570356463871244375101962759, 8.912939053555486145167316347377, 9.678230721220770104624764288567, 9.887388423961949444985083610234
