| L(s) = 1 | − 2·2-s + 4-s − 4·11-s + 16-s − 4·17-s + 8·22-s + 8·23-s − 2·25-s − 4·29-s + 8·31-s + 2·32-s + 8·34-s + 4·37-s + 16·41-s + 8·43-s − 4·44-s − 16·46-s − 12·47-s − 6·49-s + 4·50-s + 4·53-s + 8·58-s + 4·59-s + 4·61-s − 16·62-s − 11·64-s − 8·67-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 1.20·11-s + 1/4·16-s − 0.970·17-s + 1.70·22-s + 1.66·23-s − 2/5·25-s − 0.742·29-s + 1.43·31-s + 0.353·32-s + 1.37·34-s + 0.657·37-s + 2.49·41-s + 1.21·43-s − 0.603·44-s − 2.35·46-s − 1.75·47-s − 6/7·49-s + 0.565·50-s + 0.549·53-s + 1.05·58-s + 0.520·59-s + 0.512·61-s − 2.03·62-s − 1.37·64-s − 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6002963593\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6002963593\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 16 T + 138 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 24 T + 314 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 166 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.383427021460102407093609518027, −9.353210459151241721163483888227, −8.845823712072260614020394633224, −8.688259222183833575182726931236, −7.938209821736020500333671938291, −7.928715437054607046535333197668, −7.47639678854518691721782362895, −7.06720480429360381235844814450, −6.36468669123078707658828234681, −6.22162809914021680799341923990, −5.62536092260477728866504768471, −5.10655432322000120827361688973, −4.61732307188731810326957189766, −4.34033704533174739921121340134, −3.59357135158508647787518783510, −2.76494285350949537120585963927, −2.73021078120863625855221750502, −1.94348077187133837272790771925, −1.02332194949292890214877486170, −0.47797758277069428743367883195,
0.47797758277069428743367883195, 1.02332194949292890214877486170, 1.94348077187133837272790771925, 2.73021078120863625855221750502, 2.76494285350949537120585963927, 3.59357135158508647787518783510, 4.34033704533174739921121340134, 4.61732307188731810326957189766, 5.10655432322000120827361688973, 5.62536092260477728866504768471, 6.22162809914021680799341923990, 6.36468669123078707658828234681, 7.06720480429360381235844814450, 7.47639678854518691721782362895, 7.928715437054607046535333197668, 7.938209821736020500333671938291, 8.688259222183833575182726931236, 8.845823712072260614020394633224, 9.353210459151241721163483888227, 9.383427021460102407093609518027
