| L(s) = 1 | + 14·5-s − 4·11-s − 54·13-s − 14·17-s − 92·19-s + 152·23-s + 71·25-s + 106·29-s + 144·31-s + 158·37-s − 390·41-s − 508·43-s − 528·47-s − 606·53-s − 56·55-s − 364·59-s − 678·61-s − 756·65-s + 844·67-s + 8·71-s + 422·73-s + 384·79-s − 548·83-s − 196·85-s + 1.19e3·89-s − 1.28e3·95-s + 1.50e3·97-s + ⋯ |
| L(s) = 1 | + 1.25·5-s − 0.109·11-s − 1.15·13-s − 0.199·17-s − 1.11·19-s + 1.37·23-s + 0.567·25-s + 0.678·29-s + 0.834·31-s + 0.702·37-s − 1.48·41-s − 1.80·43-s − 1.63·47-s − 1.57·53-s − 0.137·55-s − 0.803·59-s − 1.42·61-s − 1.44·65-s + 1.53·67-s + 0.0133·71-s + 0.676·73-s + 0.546·79-s − 0.724·83-s − 0.250·85-s + 1.42·89-s − 1.39·95-s + 1.57·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 14 T + p^{3} T^{2} \) |
| 11 | \( 1 + 4 T + p^{3} T^{2} \) |
| 13 | \( 1 + 54 T + p^{3} T^{2} \) |
| 17 | \( 1 + 14 T + p^{3} T^{2} \) |
| 19 | \( 1 + 92 T + p^{3} T^{2} \) |
| 23 | \( 1 - 152 T + p^{3} T^{2} \) |
| 29 | \( 1 - 106 T + p^{3} T^{2} \) |
| 31 | \( 1 - 144 T + p^{3} T^{2} \) |
| 37 | \( 1 - 158 T + p^{3} T^{2} \) |
| 41 | \( 1 + 390 T + p^{3} T^{2} \) |
| 43 | \( 1 + 508 T + p^{3} T^{2} \) |
| 47 | \( 1 + 528 T + p^{3} T^{2} \) |
| 53 | \( 1 + 606 T + p^{3} T^{2} \) |
| 59 | \( 1 + 364 T + p^{3} T^{2} \) |
| 61 | \( 1 + 678 T + p^{3} T^{2} \) |
| 67 | \( 1 - 844 T + p^{3} T^{2} \) |
| 71 | \( 1 - 8 T + p^{3} T^{2} \) |
| 73 | \( 1 - 422 T + p^{3} T^{2} \) |
| 79 | \( 1 - 384 T + p^{3} T^{2} \) |
| 83 | \( 1 + 548 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1194 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1502 T + p^{3} T^{2} \) |
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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
−8.614467037088178232126517842165, −7.81935535784415108227138387633, −6.59279040186920272429682738196, −6.41038534571295420806227146525, −5.04589330251369409521972065992, −4.79236387925037506460691787587, −3.22268254652689818743861051328, −2.35862291006816767023424067526, −1.48069920034264539280955165435, 0,
1.48069920034264539280955165435, 2.35862291006816767023424067526, 3.22268254652689818743861051328, 4.79236387925037506460691787587, 5.04589330251369409521972065992, 6.41038534571295420806227146525, 6.59279040186920272429682738196, 7.81935535784415108227138387633, 8.614467037088178232126517842165
