| L(s) = 1 | − 4-s + 16-s + 14·19-s + 12·29-s − 20·31-s + 18·41-s + 10·49-s + 18·59-s − 8·61-s − 64-s + 12·71-s − 14·76-s − 4·79-s − 30·89-s + 24·101-s − 4·109-s − 12·116-s − 22·121-s + 20·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1/4·16-s + 3.21·19-s + 2.22·29-s − 3.59·31-s + 2.81·41-s + 10/7·49-s + 2.34·59-s − 1.02·61-s − 1/8·64-s + 1.42·71-s − 1.60·76-s − 0.450·79-s − 3.17·89-s + 2.38·101-s − 0.383·109-s − 1.11·116-s − 2·121-s + 1.79·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16402500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16402500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.864301452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.864301452\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 95 T^{2} + p^{2} T^{4} \) |
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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
−8.805856177567859002070517205193, −8.238351007039741974527337009330, −7.69902753231507802638820379407, −7.57722258358808735598727363065, −7.31226451940302336730862685322, −6.86909677353774157660711830655, −6.50318940822930695702618271278, −5.70264484843409895762071121840, −5.69807111918314342837917991461, −5.28887201959665023928439923614, −5.07810895614880171917144638683, −4.34608489938324046090043032395, −4.09099822425226783812750964769, −3.60672087724269147412590314271, −3.25571159188896679213064215776, −2.70865202978629777466092573965, −2.38779048958888832617022952129, −1.53438325158638567969320821890, −1.04347743841135907048745303464, −0.59433246210078581894413624233,
0.59433246210078581894413624233, 1.04347743841135907048745303464, 1.53438325158638567969320821890, 2.38779048958888832617022952129, 2.70865202978629777466092573965, 3.25571159188896679213064215776, 3.60672087724269147412590314271, 4.09099822425226783812750964769, 4.34608489938324046090043032395, 5.07810895614880171917144638683, 5.28887201959665023928439923614, 5.69807111918314342837917991461, 5.70264484843409895762071121840, 6.50318940822930695702618271278, 6.86909677353774157660711830655, 7.31226451940302336730862685322, 7.57722258358808735598727363065, 7.69902753231507802638820379407, 8.238351007039741974527337009330, 8.805856177567859002070517205193
