| L(s) = 1 | + 4·5-s + 9-s + 12·13-s + 2·17-s + 2·25-s − 12·29-s + 20·37-s − 12·41-s + 4·45-s + 2·49-s + 12·53-s − 28·61-s + 48·65-s + 20·73-s + 81-s + 8·85-s − 12·89-s − 12·97-s − 20·101-s + 20·109-s − 12·113-s + 12·117-s − 6·121-s − 28·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 1/3·9-s + 3.32·13-s + 0.485·17-s + 2/5·25-s − 2.22·29-s + 3.28·37-s − 1.87·41-s + 0.596·45-s + 2/7·49-s + 1.64·53-s − 3.58·61-s + 5.95·65-s + 2.34·73-s + 1/9·81-s + 0.867·85-s − 1.27·89-s − 1.21·97-s − 1.99·101-s + 1.91·109-s − 1.12·113-s + 1.10·117-s − 0.545·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.237600092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.237600092\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−8.820727308058655873142263738265, −8.396050359395389905793137056585, −7.86081261581705252084118405884, −7.47440021870706367601671353104, −6.59291711794167281790003444711, −6.26895088903419553976746164792, −5.91991403599991886044725193544, −5.67636732029647007076099939422, −5.13001221098195863338015289116, −4.02556354189723006943550620116, −3.94512410946611821372106342064, −3.18935016627371752862222737803, −2.35058397621900011908829435300, −1.55655451926421595753779072230, −1.26926232847077019067068429617,
1.26926232847077019067068429617, 1.55655451926421595753779072230, 2.35058397621900011908829435300, 3.18935016627371752862222737803, 3.94512410946611821372106342064, 4.02556354189723006943550620116, 5.13001221098195863338015289116, 5.67636732029647007076099939422, 5.91991403599991886044725193544, 6.26895088903419553976746164792, 6.59291711794167281790003444711, 7.47440021870706367601671353104, 7.86081261581705252084118405884, 8.396050359395389905793137056585, 8.820727308058655873142263738265
