| L(s) = 1 | − 4·17-s − 16·23-s + 6·25-s + 16·31-s − 12·41-s − 14·49-s + 16·71-s − 20·73-s − 16·79-s − 12·89-s + 4·97-s − 32·103-s − 36·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 0.970·17-s − 3.33·23-s + 6/5·25-s + 2.87·31-s − 1.87·41-s − 2·49-s + 1.89·71-s − 2.34·73-s − 1.80·79-s − 1.27·89-s + 0.406·97-s − 3.15·103-s − 3.38·113-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9472514195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9472514195\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.293519915764770492117794630436, −8.607220065156048829403997413640, −8.356040830883049492056128192654, −8.094825131228622968700602592412, −7.938371580462713834192878948760, −7.20838689010361996648963147661, −6.68218875941622221512850966157, −6.42217389574043078852694510307, −6.39646513753830288613233671949, −5.48503749812980556220380432607, −5.45088594781412998577898714603, −4.63388429120213377479475858963, −4.37885171867923715420089821869, −4.09953842333755903220408327640, −3.46762236550596602135600270505, −2.71995460686128039692108259734, −2.69099411753690930833341423639, −1.74000517821645087039298278051, −1.47608256025237012465449908004, −0.32496190451294687638815284237,
0.32496190451294687638815284237, 1.47608256025237012465449908004, 1.74000517821645087039298278051, 2.69099411753690930833341423639, 2.71995460686128039692108259734, 3.46762236550596602135600270505, 4.09953842333755903220408327640, 4.37885171867923715420089821869, 4.63388429120213377479475858963, 5.45088594781412998577898714603, 5.48503749812980556220380432607, 6.39646513753830288613233671949, 6.42217389574043078852694510307, 6.68218875941622221512850966157, 7.20838689010361996648963147661, 7.938371580462713834192878948760, 8.094825131228622968700602592412, 8.356040830883049492056128192654, 8.607220065156048829403997413640, 9.293519915764770492117794630436
