| L(s) = 1 | − 2·3-s + 3·9-s − 8·13-s − 6·17-s + 2·19-s + 5·25-s − 10·27-s − 12·29-s − 4·31-s − 2·37-s + 16·39-s + 12·41-s − 16·43-s − 12·47-s + 12·51-s − 6·53-s − 4·57-s − 6·59-s − 8·61-s − 4·67-s − 2·73-s − 10·75-s + 8·79-s + 20·81-s + 12·83-s + 24·87-s + 6·89-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s − 2.21·13-s − 1.45·17-s + 0.458·19-s + 25-s − 1.92·27-s − 2.22·29-s − 0.718·31-s − 0.328·37-s + 2.56·39-s + 1.87·41-s − 2.43·43-s − 1.75·47-s + 1.68·51-s − 0.824·53-s − 0.529·57-s − 0.781·59-s − 1.02·61-s − 0.488·67-s − 0.234·73-s − 1.15·75-s + 0.900·79-s + 20/9·81-s + 1.31·83-s + 2.57·87-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2509895752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2509895752\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−10.61662441845157271609312247549, −10.04889135363310651345323215390, −9.659057556386878572095815492825, −9.244186465240802806910759944490, −9.192650234681562265281981519467, −8.268691822816116356781575522813, −7.77589947345195472371491411327, −7.25508198704575697242647246701, −7.22149221601780025571692144702, −6.51790002346906792265498944020, −6.20316109683639666590248107495, −5.44902148242624285052927347525, −5.25573868168774993405313395900, −4.60141690972574425880329641702, −4.50458714447059805024772520608, −3.57393654806864435044477955707, −3.03705038180725135820462610431, −2.03644058740327689306834047086, −1.79666454082796095718830451387, −0.25674874227866546480281024802,
0.25674874227866546480281024802, 1.79666454082796095718830451387, 2.03644058740327689306834047086, 3.03705038180725135820462610431, 3.57393654806864435044477955707, 4.50458714447059805024772520608, 4.60141690972574425880329641702, 5.25573868168774993405313395900, 5.44902148242624285052927347525, 6.20316109683639666590248107495, 6.51790002346906792265498944020, 7.22149221601780025571692144702, 7.25508198704575697242647246701, 7.77589947345195472371491411327, 8.268691822816116356781575522813, 9.192650234681562265281981519467, 9.244186465240802806910759944490, 9.659057556386878572095815492825, 10.04889135363310651345323215390, 10.61662441845157271609312247549
