| L(s) = 1 | − 5-s − 2·7-s + 5·17-s + 4·19-s + 23-s + 25-s + 3·31-s + 2·35-s − 2·37-s − 2·41-s + 8·43-s − 9·47-s − 3·49-s + 5·53-s + 4·59-s − 7·61-s − 6·67-s + 8·71-s − 5·79-s − 16·83-s − 5·85-s + 2·89-s − 4·95-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s + 1.21·17-s + 0.917·19-s + 0.208·23-s + 1/5·25-s + 0.538·31-s + 0.338·35-s − 0.328·37-s − 0.312·41-s + 1.21·43-s − 1.31·47-s − 3/7·49-s + 0.686·53-s + 0.520·59-s − 0.896·61-s − 0.733·67-s + 0.949·71-s − 0.562·79-s − 1.75·83-s − 0.542·85-s + 0.211·89-s − 0.410·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 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
−14.15197753954443, −13.65655754015229, −13.15002474171146, −12.54371327765979, −12.29626767421124, −11.63652493724520, −11.34974032041682, −10.60839806635388, −10.06612348022804, −9.770340793835392, −9.177430447989081, −8.658199773824995, −7.956845699259729, −7.662342994680858, −6.994976197854226, −6.595842053384813, −5.838433977242387, −5.474107648374341, −4.783850147211820, −4.181335949815693, −3.414406720292286, −3.177778975806543, −2.498394647146782, −1.494356053268113, −0.8820811488583152, 0,
0.8820811488583152, 1.494356053268113, 2.498394647146782, 3.177778975806543, 3.414406720292286, 4.181335949815693, 4.783850147211820, 5.474107648374341, 5.838433977242387, 6.595842053384813, 6.994976197854226, 7.662342994680858, 7.956845699259729, 8.658199773824995, 9.177430447989081, 9.770340793835392, 10.06612348022804, 10.60839806635388, 11.34974032041682, 11.63652493724520, 12.29626767421124, 12.54371327765979, 13.15002474171146, 13.65655754015229, 14.15197753954443
