| L(s) = 1 | − 2·2-s + 3·4-s − 2·5-s − 2·7-s − 4·8-s + 4·10-s − 2·11-s + 4·13-s + 4·14-s + 5·16-s − 6·17-s + 8·19-s − 6·20-s + 4·22-s − 2·23-s + 3·25-s − 8·26-s − 6·28-s − 4·29-s − 6·32-s + 12·34-s + 4·35-s − 6·37-s − 16·38-s + 8·40-s − 4·41-s − 6·44-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.894·5-s − 0.755·7-s − 1.41·8-s + 1.26·10-s − 0.603·11-s + 1.10·13-s + 1.06·14-s + 5/4·16-s − 1.45·17-s + 1.83·19-s − 1.34·20-s + 0.852·22-s − 0.417·23-s + 3/5·25-s − 1.56·26-s − 1.13·28-s − 0.742·29-s − 1.06·32-s + 2.05·34-s + 0.676·35-s − 0.986·37-s − 2.59·38-s + 1.26·40-s − 0.624·41-s − 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4284900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4284900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6111326853\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6111326853\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 10 T + 130 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 162 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−9.072268106338669039789939427855, −9.001689272658231811437568414186, −8.577001114223972297420009252660, −8.285017207846478270124164635711, −7.75620197222329771824124911012, −7.45048294540669792161821743024, −6.99559966254375796342627411681, −6.96847314861357414786976546877, −6.14012706877911246905249672678, −6.06916702646369288123974496123, −5.31607040833313925434530722999, −5.13766739098388302850912152623, −4.16008454099263464355734596640, −3.96803155340569292580337095906, −3.25763803006833731045864181165, −3.11439405559670064525366173300, −2.32612613328561852707480352135, −1.86844773631523649823894922866, −0.980755982806687577424829286357, −0.43167941363320819501824277025,
0.43167941363320819501824277025, 0.980755982806687577424829286357, 1.86844773631523649823894922866, 2.32612613328561852707480352135, 3.11439405559670064525366173300, 3.25763803006833731045864181165, 3.96803155340569292580337095906, 4.16008454099263464355734596640, 5.13766739098388302850912152623, 5.31607040833313925434530722999, 6.06916702646369288123974496123, 6.14012706877911246905249672678, 6.96847314861357414786976546877, 6.99559966254375796342627411681, 7.45048294540669792161821743024, 7.75620197222329771824124911012, 8.285017207846478270124164635711, 8.577001114223972297420009252660, 9.001689272658231811437568414186, 9.072268106338669039789939427855
