| L(s) = 1 | − 2-s − 2·7-s − 8-s + 6·13-s + 2·14-s − 16-s − 9·17-s − 6·19-s + 9·23-s + 3·25-s − 6·26-s − 13·29-s + 2·31-s + 6·32-s + 9·34-s + 4·37-s + 6·38-s − 14·41-s + 7·43-s − 9·46-s − 7·47-s + 3·49-s − 3·50-s + 15·53-s + 2·56-s + 13·58-s − 17·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.755·7-s − 0.353·8-s + 1.66·13-s + 0.534·14-s − 1/4·16-s − 2.18·17-s − 1.37·19-s + 1.87·23-s + 3/5·25-s − 1.17·26-s − 2.41·29-s + 0.359·31-s + 1.06·32-s + 1.54·34-s + 0.657·37-s + 0.973·38-s − 2.18·41-s + 1.06·43-s − 1.32·46-s − 1.02·47-s + 3/7·49-s − 0.424·50-s + 2.06·53-s + 0.267·56-s + 1.70·58-s − 2.21·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 9 T + 3 p T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 9 T + 63 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 13 T + 97 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 7 T + 95 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 7 T + 77 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 15 T + 133 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 17 T + 187 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 125 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T + 53 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 145 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 13 T + 197 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 3 T - 83 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 181 T^{2} + p^{2} T^{4} \) |
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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
−7.77758870980026191443477923205, −7.19917712167146028912922421595, −6.95244404217706450358545438050, −6.67213960577919432482312127610, −6.50799480124614801145048774027, −6.06153897662368754295501234348, −5.67547434115396659549410619232, −5.42893754340545760528481747431, −4.73257386284367800960015847693, −4.43732231556913125379368389792, −4.13266444564653724124166864279, −3.80673019137258722371733740744, −3.08758312299516695693876691755, −3.02601891830972399067159328603, −2.47757018947875371199147973196, −1.89080315756901568626900252622, −1.55294342342750615369647238771, −0.880751857162618830403849390093, 0, 0,
0.880751857162618830403849390093, 1.55294342342750615369647238771, 1.89080315756901568626900252622, 2.47757018947875371199147973196, 3.02601891830972399067159328603, 3.08758312299516695693876691755, 3.80673019137258722371733740744, 4.13266444564653724124166864279, 4.43732231556913125379368389792, 4.73257386284367800960015847693, 5.42893754340545760528481747431, 5.67547434115396659549410619232, 6.06153897662368754295501234348, 6.50799480124614801145048774027, 6.67213960577919432482312127610, 6.95244404217706450358545438050, 7.19917712167146028912922421595, 7.77758870980026191443477923205
