| L(s) = 1 | − 9-s − 25-s + 20·31-s − 12·41-s − 14·49-s − 16·71-s − 28·73-s + 4·79-s + 81-s − 28·89-s − 12·97-s + 8·103-s + 16·113-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 1/5·25-s + 3.59·31-s − 1.87·41-s − 2·49-s − 1.89·71-s − 3.27·73-s + 0.450·79-s + 1/9·81-s − 2.96·89-s − 1.21·97-s + 0.788·103-s + 1.50·113-s + 1.63·121-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 + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.359269910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.359269910\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 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
−8.827799990712977520946451650733, −8.254416835610381232494810699841, −8.186520092661786037269178760602, −7.50486938417885545124055800620, −7.27298773460456112701357867454, −6.70834588821913602128006958058, −6.53982359712344958064973030090, −6.08100528540742446689186733800, −5.74721036129320381496771524872, −5.37860504503832044374770254001, −4.70372321028102399566329895097, −4.45451497077767334482364600460, −4.41115958037146573701622686550, −3.42159874866283560332052222849, −3.23284594210212204110614590050, −2.78259383438803371211110926210, −2.37517907771384770238424012251, −1.52985034813097404686160527484, −1.31461538318645763983816791281, −0.34642160704159405128224333288,
0.34642160704159405128224333288, 1.31461538318645763983816791281, 1.52985034813097404686160527484, 2.37517907771384770238424012251, 2.78259383438803371211110926210, 3.23284594210212204110614590050, 3.42159874866283560332052222849, 4.41115958037146573701622686550, 4.45451497077767334482364600460, 4.70372321028102399566329895097, 5.37860504503832044374770254001, 5.74721036129320381496771524872, 6.08100528540742446689186733800, 6.53982359712344958064973030090, 6.70834588821913602128006958058, 7.27298773460456112701357867454, 7.50486938417885545124055800620, 8.186520092661786037269178760602, 8.254416835610381232494810699841, 8.827799990712977520946451650733
