| L(s) = 1 | − 5·9-s − 4·11-s − 4·19-s − 36·29-s − 8·31-s + 4·41-s − 11·49-s − 20·59-s − 18·61-s + 56·71-s + 28·79-s + 9·81-s − 30·89-s + 20·99-s − 6·101-s − 2·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + ⋯ |
| L(s) = 1 | − 5/3·9-s − 1.20·11-s − 0.917·19-s − 6.68·29-s − 1.43·31-s + 0.624·41-s − 1.57·49-s − 2.60·59-s − 2.30·61-s + 6.64·71-s + 3.15·79-s + 81-s − 3.17·89-s + 2.01·99-s − 0.597·101-s − 0.191·109-s + 2.36·121-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 + 4·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5071427996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5071427996\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2^3$ | \( 1 + 5 T^{2} + 16 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 + 18 T^{2} + 35 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 45 T^{2} + 1496 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 58 T^{2} + 1995 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 + 6 T^{2} - 2773 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 9 T + 20 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 122 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 2 T^{2} - 5325 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 14 T + 117 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 45 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.96411468009677848550885179092, −6.45192933827419956587209055068, −6.41358048075850871332563902085, −6.05563665250773354964402971804, −5.83179161131033372613572089833, −5.73011305717528525436359989637, −5.63949061459481675727639200303, −5.31430870361603401102157181646, −5.02516477771015952698130739756, −4.91319735809698487200322051045, −4.88760439311596312604560381989, −4.22494646622992729183394093757, −4.00272399517110502234880523777, −3.73283270797228823878172899156, −3.66271441719513265454371565313, −3.35922544540925577761682698651, −3.23922381633568738896739154300, −2.66158356379609903894073574869, −2.62835528059228636525604196630, −2.05511623152055733804699600409, −1.94066432002736558143614523660, −1.87203923973009453159345560663, −1.41421674611134668641749669216, −0.37116713737588800641613554818, −0.29255729232755309651972439819,
0.29255729232755309651972439819, 0.37116713737588800641613554818, 1.41421674611134668641749669216, 1.87203923973009453159345560663, 1.94066432002736558143614523660, 2.05511623152055733804699600409, 2.62835528059228636525604196630, 2.66158356379609903894073574869, 3.23922381633568738896739154300, 3.35922544540925577761682698651, 3.66271441719513265454371565313, 3.73283270797228823878172899156, 4.00272399517110502234880523777, 4.22494646622992729183394093757, 4.88760439311596312604560381989, 4.91319735809698487200322051045, 5.02516477771015952698130739756, 5.31430870361603401102157181646, 5.63949061459481675727639200303, 5.73011305717528525436359989637, 5.83179161131033372613572089833, 6.05563665250773354964402971804, 6.41358048075850871332563902085, 6.45192933827419956587209055068, 6.96411468009677848550885179092
