| L(s) = 1 | + 2·3-s + 3·7-s + 9-s − 11-s − 5·17-s + 4·19-s + 6·21-s − 2·23-s − 4·27-s − 3·29-s − 5·31-s − 2·33-s + 6·37-s + 8·41-s − 6·43-s + 47-s + 2·49-s − 10·51-s + 11·53-s + 8·57-s − 3·59-s − 5·61-s + 3·63-s − 9·67-s − 4·69-s − 8·71-s + 2·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.13·7-s + 1/3·9-s − 0.301·11-s − 1.21·17-s + 0.917·19-s + 1.30·21-s − 0.417·23-s − 0.769·27-s − 0.557·29-s − 0.898·31-s − 0.348·33-s + 0.986·37-s + 1.24·41-s − 0.914·43-s + 0.145·47-s + 2/7·49-s − 1.40·51-s + 1.51·53-s + 1.05·57-s − 0.390·59-s − 0.640·61-s + 0.377·63-s − 1.09·67-s − 0.481·69-s − 0.949·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−14.97039050524858, −14.76621843544302, −14.31586850848030, −13.60939903638911, −13.39846412676675, −12.84687301841962, −11.91869401148762, −11.63280104939915, −10.91829270834656, −10.62210172065613, −9.671774534342431, −9.258027214856256, −8.800791624285229, −8.186486809608838, −7.752216815051639, −7.357122871103668, −6.596109034851919, −5.680775839025619, −5.318726580952952, −4.358927373606802, −4.096065730000094, −3.161356866190285, −2.569545385405993, −1.966225591293810, −1.296826175582289, 0,
1.296826175582289, 1.966225591293810, 2.569545385405993, 3.161356866190285, 4.096065730000094, 4.358927373606802, 5.318726580952952, 5.680775839025619, 6.596109034851919, 7.357122871103668, 7.752216815051639, 8.186486809608838, 8.800791624285229, 9.258027214856256, 9.671774534342431, 10.62210172065613, 10.91829270834656, 11.63280104939915, 11.91869401148762, 12.84687301841962, 13.39846412676675, 13.60939903638911, 14.31586850848030, 14.76621843544302, 14.97039050524858
