L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s + 4·11-s + 13-s + 2·14-s + 16-s + 4·17-s + 6·19-s − 4·22-s − 26-s − 2·28-s − 4·29-s + 6·31-s − 32-s − 4·34-s + 2·37-s − 6·38-s + 10·41-s − 8·43-s + 4·44-s − 3·49-s + 52-s + 4·53-s + 2·56-s + 4·58-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s + 1.20·11-s + 0.277·13-s + 0.534·14-s + 1/4·16-s + 0.970·17-s + 1.37·19-s − 0.852·22-s − 0.196·26-s − 0.377·28-s − 0.742·29-s + 1.07·31-s − 0.176·32-s − 0.685·34-s + 0.328·37-s − 0.973·38-s + 1.56·41-s − 1.21·43-s + 0.603·44-s − 3/7·49-s + 0.138·52-s + 0.549·53-s + 0.267·56-s + 0.525·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.528847391\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.528847391\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−8.102305030334385717759653693420, −7.42189798987211103301752874167, −6.78786542092977440414998354579, −6.08031988046800902723224040992, −5.49030414174518463418405201881, −4.31926070867023278690169395990, −3.45785833421138596271514115336, −2.87439747255041218267121264549, −1.56783789094817932590431935196, −0.78127937328209651773208440913,
0.78127937328209651773208440913, 1.56783789094817932590431935196, 2.87439747255041218267121264549, 3.45785833421138596271514115336, 4.31926070867023278690169395990, 5.49030414174518463418405201881, 6.08031988046800902723224040992, 6.78786542092977440414998354579, 7.42189798987211103301752874167, 8.102305030334385717759653693420