| L(s) = 1 | + 2-s + 2·4-s + 2·5-s − 5·7-s + 5·8-s + 2·10-s − 3·11-s − 2·13-s − 5·14-s + 5·16-s − 2·17-s + 5·19-s + 4·20-s − 3·22-s − 6·23-s − 7·25-s − 2·26-s − 10·28-s − 18·29-s + 8·31-s + 10·32-s − 2·34-s − 10·35-s − 8·37-s + 5·38-s + 10·40-s + 18·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 4-s + 0.894·5-s − 1.88·7-s + 1.76·8-s + 0.632·10-s − 0.904·11-s − 0.554·13-s − 1.33·14-s + 5/4·16-s − 0.485·17-s + 1.14·19-s + 0.894·20-s − 0.639·22-s − 1.25·23-s − 7/5·25-s − 0.392·26-s − 1.88·28-s − 3.34·29-s + 1.43·31-s + 1.76·32-s − 0.342·34-s − 1.69·35-s − 1.31·37-s + 0.811·38-s + 1.58·40-s + 2.74·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3956121 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3956121 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.735325164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.735325164\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 + 5 T + 15 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 39 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 57 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 69 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 74 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 11 T + 131 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 113 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 19 T + 207 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 18 T + 194 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 141 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 153 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 145 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 9 T + 67 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 7 T + 75 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
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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
−9.409238236111293147614991943784, −9.245341108181650453864445936191, −8.635208035074631267727363154814, −7.895597361737281177936433490654, −7.58688030916889068911840888758, −7.37077881772501712470701174459, −7.09837761815640573339823309884, −6.42513051436587893262599631917, −6.15371470161976494883295109400, −5.78028982566473091361388393580, −5.39166931605969474855862588523, −5.22120476527112420991759421370, −4.16815659854820814504846634635, −4.13187737108081911914862773049, −3.58245666235853924064529119159, −2.97496889118083085041602172832, −2.48250622083927589940322463339, −2.10179275559520304299589898900, −1.68295121708550894339422473691, −0.48003317744982964941851659877,
0.48003317744982964941851659877, 1.68295121708550894339422473691, 2.10179275559520304299589898900, 2.48250622083927589940322463339, 2.97496889118083085041602172832, 3.58245666235853924064529119159, 4.13187737108081911914862773049, 4.16815659854820814504846634635, 5.22120476527112420991759421370, 5.39166931605969474855862588523, 5.78028982566473091361388393580, 6.15371470161976494883295109400, 6.42513051436587893262599631917, 7.09837761815640573339823309884, 7.37077881772501712470701174459, 7.58688030916889068911840888758, 7.895597361737281177936433490654, 8.635208035074631267727363154814, 9.245341108181650453864445936191, 9.409238236111293147614991943784
