L(s) = 1 | − 2·2-s − 2·3-s + 2·4-s − 4·5-s + 4·6-s − 3·7-s − 4·8-s + 8·10-s − 2·11-s − 4·12-s − 7·13-s + 6·14-s + 8·15-s + 8·16-s − 5·17-s + 3·19-s − 8·20-s + 6·21-s + 4·22-s + 4·23-s + 8·24-s + 8·25-s + 14·26-s + 2·27-s − 6·28-s − 2·29-s − 16·30-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 1.15·3-s + 4-s − 1.78·5-s + 1.63·6-s − 1.13·7-s − 1.41·8-s + 2.52·10-s − 0.603·11-s − 1.15·12-s − 1.94·13-s + 1.60·14-s + 2.06·15-s + 2·16-s − 1.21·17-s + 0.688·19-s − 1.78·20-s + 1.30·21-s + 0.852·22-s + 0.834·23-s + 1.63·24-s + 8/5·25-s + 2.74·26-s + 0.384·27-s − 1.13·28-s − 0.371·29-s − 2.92·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14545 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14545 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 3 T + p T^{2} ) \) |
| 2909 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 54 T + p T^{2} ) \) |
good | 2 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 7 T + 29 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + p T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 44 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 48 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 11 T + 87 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 9 T + 130 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 33 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 5 T + 58 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 88 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 3 T - 78 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 3 T - 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 15 T + 195 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T - 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 11 T + 180 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
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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
−16.7805060903, −16.2080799492, −15.7069171143, −15.4658448368, −15.0699853630, −14.5118113261, −13.5419714339, −12.9155259904, −12.2223050381, −12.0440288691, −11.7736749488, −11.0374928601, −10.7357295436, −9.94103551770, −9.57005397486, −9.05035777134, −8.31237684949, −7.89385212454, −7.25251504060, −6.70783988078, −6.20559277636, −5.16110088170, −4.71338274182, −3.36146802901, −2.83831539222, 0, 0,
2.83831539222, 3.36146802901, 4.71338274182, 5.16110088170, 6.20559277636, 6.70783988078, 7.25251504060, 7.89385212454, 8.31237684949, 9.05035777134, 9.57005397486, 9.94103551770, 10.7357295436, 11.0374928601, 11.7736749488, 12.0440288691, 12.2223050381, 12.9155259904, 13.5419714339, 14.5118113261, 15.0699853630, 15.4658448368, 15.7069171143, 16.2080799492, 16.7805060903