| L(s) = 1 | − 3-s + 2·7-s + 9-s + 12·13-s + 8·19-s − 2·21-s − 6·25-s − 27-s − 16·31-s + 12·37-s − 12·39-s − 8·43-s + 3·49-s − 8·57-s − 4·61-s + 2·63-s + 24·67-s − 28·73-s + 6·75-s − 16·79-s + 81-s + 24·91-s + 16·93-s + 20·97-s − 4·109-s − 12·111-s + 12·117-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s + 3.32·13-s + 1.83·19-s − 0.436·21-s − 6/5·25-s − 0.192·27-s − 2.87·31-s + 1.97·37-s − 1.92·39-s − 1.21·43-s + 3/7·49-s − 1.05·57-s − 0.512·61-s + 0.251·63-s + 2.93·67-s − 3.27·73-s + 0.692·75-s − 1.80·79-s + 1/9·81-s + 2.51·91-s + 1.65·93-s + 2.03·97-s − 0.383·109-s − 1.13·111-s + 1.10·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84672 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84672 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.606017398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.606017398\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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
−9.826688574335115321087905060003, −8.965525895765181533711100026451, −8.859066757414730662597301630531, −8.209746531513691341164468151877, −7.51563481619318596865649774866, −7.43966046425964616959399153516, −6.33141366174552935025595412995, −6.14074423707739109326089273460, −5.47137535045602222376296375819, −5.22329793878816306895756424483, −4.05565358957946867217249156509, −3.85282553104851107098279692910, −3.14075748465669101106954045259, −1.72065991354884586405445664376, −1.18803818723015611426762577075,
1.18803818723015611426762577075, 1.72065991354884586405445664376, 3.14075748465669101106954045259, 3.85282553104851107098279692910, 4.05565358957946867217249156509, 5.22329793878816306895756424483, 5.47137535045602222376296375819, 6.14074423707739109326089273460, 6.33141366174552935025595412995, 7.43966046425964616959399153516, 7.51563481619318596865649774866, 8.209746531513691341164468151877, 8.859066757414730662597301630531, 8.965525895765181533711100026451, 9.826688574335115321087905060003
