| L(s) = 1 | + 2·3-s − 2·7-s + 3·9-s + 11-s + 4·13-s − 2·17-s + 9·19-s − 4·21-s − 2·23-s − 10·25-s + 4·27-s + 4·29-s + 6·31-s + 2·33-s − 6·37-s + 8·39-s − 41-s − 10·43-s + 19·47-s + 3·49-s − 4·51-s + 53-s + 18·57-s + 9·59-s + 19·61-s − 6·63-s + 14·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.755·7-s + 9-s + 0.301·11-s + 1.10·13-s − 0.485·17-s + 2.06·19-s − 0.872·21-s − 0.417·23-s − 2·25-s + 0.769·27-s + 0.742·29-s + 1.07·31-s + 0.348·33-s − 0.986·37-s + 1.28·39-s − 0.156·41-s − 1.52·43-s + 2.77·47-s + 3/7·49-s − 0.560·51-s + 0.137·53-s + 2.38·57-s + 1.17·59-s + 2.43·61-s − 0.755·63-s + 1.71·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59721984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59721984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.169158881\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.169158881\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - T + 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 9 T + 48 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 42 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + T + 72 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 19 T + 174 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - T + 14 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 9 T + 46 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 19 T + 202 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 14 T + 142 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 106 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 142 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−8.069616465114074625420058004004, −7.76955891031587878766423839136, −7.34874015662898059626013642731, −7.03156084083895790518773447618, −6.69526370235415051291744972340, −6.47621867360384587092488930658, −5.82871814383236284895405728985, −5.78210514950836556481181390953, −5.12341442530325497758611591396, −5.02797714620924519657772209037, −4.16747302053852780862616653060, −4.00238437579212229204889708246, −3.56428201782535051303692873767, −3.55452307930884782522302475185, −2.87293166624421301343379082092, −2.55866122252602200141447400123, −2.06141308386552902979139088759, −1.67372604256424874501373601747, −0.928812032894187998649617790708, −0.65701053399393640961861305475,
0.65701053399393640961861305475, 0.928812032894187998649617790708, 1.67372604256424874501373601747, 2.06141308386552902979139088759, 2.55866122252602200141447400123, 2.87293166624421301343379082092, 3.55452307930884782522302475185, 3.56428201782535051303692873767, 4.00238437579212229204889708246, 4.16747302053852780862616653060, 5.02797714620924519657772209037, 5.12341442530325497758611591396, 5.78210514950836556481181390953, 5.82871814383236284895405728985, 6.47621867360384587092488930658, 6.69526370235415051291744972340, 7.03156084083895790518773447618, 7.34874015662898059626013642731, 7.76955891031587878766423839136, 8.069616465114074625420058004004
