| L(s) = 1 | + 2·3-s − 2·5-s + 8·7-s − 9-s + 2·11-s − 2·13-s − 4·15-s − 4·17-s − 4·19-s + 16·21-s + 4·23-s + 25-s − 6·27-s + 2·29-s + 14·31-s + 4·33-s − 16·35-s + 8·37-s − 4·39-s − 8·41-s − 6·43-s + 2·45-s + 10·47-s + 34·49-s − 8·51-s − 14·53-s − 4·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 3.02·7-s − 1/3·9-s + 0.603·11-s − 0.554·13-s − 1.03·15-s − 0.970·17-s − 0.917·19-s + 3.49·21-s + 0.834·23-s + 1/5·25-s − 1.15·27-s + 0.371·29-s + 2.51·31-s + 0.696·33-s − 2.70·35-s + 1.31·37-s − 0.640·39-s − 1.24·41-s − 0.914·43-s + 0.298·45-s + 1.45·47-s + 34/7·49-s − 1.12·51-s − 1.92·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 215296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.918329839\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.918329839\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 21 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T - 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_4$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 14 T + 109 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 93 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 50 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 194 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 162 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−11.27978742913538312169047009072, −10.96765844679583239577581789989, −10.58626164264797297566442087863, −9.875258344557879858361754604759, −9.236626649245502236479080212898, −8.741213086458248762446308351543, −8.396036528851822239321928881534, −8.274640772805611287806435115955, −7.73303275745528186431136692158, −7.61242434974079886188390912790, −6.56141388687771220556256484609, −6.44607464706864090663417722018, −5.15612823620374243376983888562, −5.04996163437703155781564337138, −4.31712109237485319624595677921, −4.23680761208627097828734807028, −3.23158565572461833716557048927, −2.47026363521582101871036277138, −2.06687702977457164520023690482, −1.10225201330208664806014390133,
1.10225201330208664806014390133, 2.06687702977457164520023690482, 2.47026363521582101871036277138, 3.23158565572461833716557048927, 4.23680761208627097828734807028, 4.31712109237485319624595677921, 5.04996163437703155781564337138, 5.15612823620374243376983888562, 6.44607464706864090663417722018, 6.56141388687771220556256484609, 7.61242434974079886188390912790, 7.73303275745528186431136692158, 8.274640772805611287806435115955, 8.396036528851822239321928881534, 8.741213086458248762446308351543, 9.236626649245502236479080212898, 9.875258344557879858361754604759, 10.58626164264797297566442087863, 10.96765844679583239577581789989, 11.27978742913538312169047009072
