| L(s) = 1 | + 2-s + 4-s + 4·7-s + 8-s − 13-s + 4·14-s + 16-s + 17-s − 4·19-s − 26-s + 4·28-s − 6·29-s − 4·31-s + 32-s + 34-s − 2·37-s − 4·38-s − 6·41-s + 4·43-s + 9·49-s − 52-s + 6·53-s + 4·56-s − 6·58-s + 12·59-s − 10·61-s − 4·62-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.51·7-s + 0.353·8-s − 0.277·13-s + 1.06·14-s + 1/4·16-s + 0.242·17-s − 0.917·19-s − 0.196·26-s + 0.755·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.171·34-s − 0.328·37-s − 0.648·38-s − 0.937·41-s + 0.609·43-s + 9/7·49-s − 0.138·52-s + 0.824·53-s + 0.534·56-s − 0.787·58-s + 1.56·59-s − 1.28·61-s − 0.508·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.677234668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.677234668\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.87097131937554, −13.20462042661724, −12.88528906210015, −12.28050317692464, −11.77565547353500, −11.43738728896667, −10.78303854742103, −10.66288037032487, −9.880371073098795, −9.302159402182834, −8.605182369823894, −8.259944467923404, −7.704833475314789, −7.157435756406275, −6.769814005141877, −5.910271290594237, −5.486750670600627, −5.049252617143708, −4.461465975409796, −3.975437096068479, −3.421040757113782, −2.536249008983743, −1.969024628615591, −1.565222437839251, −0.5791336123300648,
0.5791336123300648, 1.565222437839251, 1.969024628615591, 2.536249008983743, 3.421040757113782, 3.975437096068479, 4.461465975409796, 5.049252617143708, 5.486750670600627, 5.910271290594237, 6.769814005141877, 7.157435756406275, 7.704833475314789, 8.259944467923404, 8.605182369823894, 9.302159402182834, 9.880371073098795, 10.66288037032487, 10.78303854742103, 11.43738728896667, 11.77565547353500, 12.28050317692464, 12.88528906210015, 13.20462042661724, 13.87097131937554
