| L(s) = 1 | + 3-s + 8·7-s + 9-s + 12·13-s − 8·19-s + 8·21-s + 25-s + 27-s + 8·31-s − 20·37-s + 12·39-s + 8·43-s + 34·49-s − 8·57-s − 20·61-s + 8·63-s − 24·67-s + 20·73-s + 75-s + 8·79-s + 81-s + 96·91-s + 8·93-s − 28·97-s − 8·103-s − 4·109-s − 20·111-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 3.02·7-s + 1/3·9-s + 3.32·13-s − 1.83·19-s + 1.74·21-s + 1/5·25-s + 0.192·27-s + 1.43·31-s − 3.28·37-s + 1.92·39-s + 1.21·43-s + 34/7·49-s − 1.05·57-s − 2.56·61-s + 1.00·63-s − 2.93·67-s + 2.34·73-s + 0.115·75-s + 0.900·79-s + 1/9·81-s + 10.0·91-s + 0.829·93-s − 2.84·97-s − 0.788·103-s − 0.383·109-s − 1.89·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 691200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 691200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.470350448\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.470350448\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−8.234396477105587706249662636004, −8.116410983938513625426203426805, −7.75866687525969370360077670275, −7.03872324148136889156556818396, −6.39854886739801755961916412419, −6.21305657818228885677430783028, −5.44404884721337967389713010913, −5.11098268655570501466709405394, −4.36333484257300403943968845908, −4.23445324460423189498619158076, −3.69324260965892944900735137760, −2.94053502252576914533622368524, −2.02785893596435789244461953313, −1.47970352314852327908767006362, −1.31634970517745226225569387554,
1.31634970517745226225569387554, 1.47970352314852327908767006362, 2.02785893596435789244461953313, 2.94053502252576914533622368524, 3.69324260965892944900735137760, 4.23445324460423189498619158076, 4.36333484257300403943968845908, 5.11098268655570501466709405394, 5.44404884721337967389713010913, 6.21305657818228885677430783028, 6.39854886739801755961916412419, 7.03872324148136889156556818396, 7.75866687525969370360077670275, 8.116410983938513625426203426805, 8.234396477105587706249662636004
