| L(s) = 1 | + 2-s + 4-s + 8-s − 4·11-s − 2·13-s + 16-s − 17-s − 4·19-s − 4·22-s + 8·23-s − 2·26-s − 6·29-s + 32-s − 34-s + 2·37-s − 4·38-s + 10·41-s + 4·43-s − 4·44-s + 8·46-s − 2·52-s + 6·53-s − 6·58-s − 4·59-s − 6·61-s + 64-s + 12·67-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.20·11-s − 0.554·13-s + 1/4·16-s − 0.242·17-s − 0.917·19-s − 0.852·22-s + 1.66·23-s − 0.392·26-s − 1.11·29-s + 0.176·32-s − 0.171·34-s + 0.328·37-s − 0.648·38-s + 1.56·41-s + 0.609·43-s − 0.603·44-s + 1.17·46-s − 0.277·52-s + 0.824·53-s − 0.787·58-s − 0.520·59-s − 0.768·61-s + 1/8·64-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.939738167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.939738167\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.56264366517044, −12.33090158827420, −11.41113570235612, −11.22216398786598, −10.74562407318561, −10.45228519217585, −9.831447159180900, −9.236127164468893, −8.989621768372266, −8.263981828126081, −7.766105245281418, −7.459442724565829, −6.926964539751046, −6.469921967027341, −5.865686998857288, −5.411044323594955, −5.034015416241794, −4.510923611259118, −4.048707230819426, −3.452578360028474, −2.759532780083599, −2.486928083149086, −1.971291124837190, −1.107568151370340, −0.3991663832869653,
0.3991663832869653, 1.107568151370340, 1.971291124837190, 2.486928083149086, 2.759532780083599, 3.452578360028474, 4.048707230819426, 4.510923611259118, 5.034015416241794, 5.411044323594955, 5.865686998857288, 6.469921967027341, 6.926964539751046, 7.459442724565829, 7.766105245281418, 8.263981828126081, 8.989621768372266, 9.236127164468893, 9.831447159180900, 10.45228519217585, 10.74562407318561, 11.22216398786598, 11.41113570235612, 12.33090158827420, 12.56264366517044
