| L(s) = 1 | − 4·5-s + 9-s + 8·11-s − 4·13-s + 2·25-s + 16·31-s + 8·43-s − 4·45-s − 7·49-s − 32·55-s − 4·61-s + 16·65-s − 8·67-s + 81-s + 8·99-s − 36·101-s + 32·103-s − 24·107-s + 36·113-s − 4·117-s + 26·121-s + 28·125-s + 127-s + 131-s + 137-s + 139-s − 32·143-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1/3·9-s + 2.41·11-s − 1.10·13-s + 2/5·25-s + 2.87·31-s + 1.21·43-s − 0.596·45-s − 49-s − 4.31·55-s − 0.512·61-s + 1.98·65-s − 0.977·67-s + 1/9·81-s + 0.804·99-s − 3.58·101-s + 3.15·103-s − 2.32·107-s + 3.38·113-s − 0.369·117-s + 2.36·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.67·143-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225792 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225792 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.242914371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.242914371\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.152797891745064115144378653628, −8.363976058602769231488256790631, −8.098990694093691505068092710868, −7.56975735076810513912224977904, −7.22105516751355937830285941842, −6.54491697858797937973081677578, −6.42897107072744719896577758454, −5.63168120431032402488374725093, −4.69719753757653070842273737242, −4.25303028692796488061253338187, −4.18766143896287814919024894984, −3.41088997198904527337952221517, −2.81065807634666280216127599883, −1.69952852024005505825174252057, −0.72841599673746501614747483703,
0.72841599673746501614747483703, 1.69952852024005505825174252057, 2.81065807634666280216127599883, 3.41088997198904527337952221517, 4.18766143896287814919024894984, 4.25303028692796488061253338187, 4.69719753757653070842273737242, 5.63168120431032402488374725093, 6.42897107072744719896577758454, 6.54491697858797937973081677578, 7.22105516751355937830285941842, 7.56975735076810513912224977904, 8.098990694093691505068092710868, 8.363976058602769231488256790631, 9.152797891745064115144378653628
