| L(s) = 1 | + 2·3-s − 2·5-s − 2·7-s + 3·9-s − 4·15-s − 4·21-s − 4·23-s + 3·25-s + 4·27-s − 8·29-s + 4·31-s + 4·35-s − 8·37-s + 4·41-s + 12·43-s − 6·45-s − 4·47-s + 3·49-s − 4·53-s − 8·59-s − 16·61-s − 6·63-s − 4·67-s − 8·69-s − 4·71-s + 8·73-s + 6·75-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 0.755·7-s + 9-s − 1.03·15-s − 0.872·21-s − 0.834·23-s + 3/5·25-s + 0.769·27-s − 1.48·29-s + 0.718·31-s + 0.676·35-s − 1.31·37-s + 0.624·41-s + 1.82·43-s − 0.894·45-s − 0.583·47-s + 3/7·49-s − 0.549·53-s − 1.04·59-s − 2.04·61-s − 0.755·63-s − 0.488·67-s − 0.963·69-s − 0.474·71-s + 0.936·73-s + 0.692·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45158400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45158400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_4$ | \( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T - 34 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 174 T^{2} + p^{2} T^{4} \) |
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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
−7.71737386239970724107276566658, −7.66925481227183068715037523996, −7.06310245850453679588045455939, −7.05772603733362308708037654405, −6.39815353259391717705448874127, −6.16102038097593345748496207125, −5.73950163571921402777630422387, −5.33155746228840150640317966158, −4.80490593823448830736869991021, −4.37239236664407068355833447521, −4.07067805196829668824597279912, −3.82444055445131819105349789808, −3.21913839161527807412317866279, −3.18322185806839573975118533247, −2.45949538744361587957419265616, −2.36188033784358118824904054588, −1.38288848116453940228345975272, −1.33855236197535868931855452825, 0, 0,
1.33855236197535868931855452825, 1.38288848116453940228345975272, 2.36188033784358118824904054588, 2.45949538744361587957419265616, 3.18322185806839573975118533247, 3.21913839161527807412317866279, 3.82444055445131819105349789808, 4.07067805196829668824597279912, 4.37239236664407068355833447521, 4.80490593823448830736869991021, 5.33155746228840150640317966158, 5.73950163571921402777630422387, 6.16102038097593345748496207125, 6.39815353259391717705448874127, 7.05772603733362308708037654405, 7.06310245850453679588045455939, 7.66925481227183068715037523996, 7.71737386239970724107276566658
