| L(s) = 1 | − 4·7-s + 4·11-s − 6·13-s + 2·17-s − 4·19-s − 10·29-s + 4·31-s + 10·37-s − 2·41-s − 4·43-s − 8·47-s + 9·49-s + 2·53-s + 12·59-s − 10·61-s + 12·67-s − 10·73-s − 16·77-s + 4·79-s − 4·83-s + 6·89-s + 24·91-s + 14·97-s − 2·101-s + 4·103-s + 12·107-s − 2·109-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 1.20·11-s − 1.66·13-s + 0.485·17-s − 0.917·19-s − 1.85·29-s + 0.718·31-s + 1.64·37-s − 0.312·41-s − 0.609·43-s − 1.16·47-s + 9/7·49-s + 0.274·53-s + 1.56·59-s − 1.28·61-s + 1.46·67-s − 1.17·73-s − 1.82·77-s + 0.450·79-s − 0.439·83-s + 0.635·89-s + 2.51·91-s + 1.42·97-s − 0.199·101-s + 0.394·103-s + 1.16·107-s − 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.081313375\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.081313375\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.78145398812000971262690871554, −7.14375287833410078479431736650, −6.50633388063324819338747364862, −6.03330214742327009150557463484, −5.09910422861743261288259915465, −4.22963585877726431047642501901, −3.56169383752156022497502286286, −2.77254402767806087386047253538, −1.89785254017853770558179110971, −0.50389200920024116772639675277,
0.50389200920024116772639675277, 1.89785254017853770558179110971, 2.77254402767806087386047253538, 3.56169383752156022497502286286, 4.22963585877726431047642501901, 5.09910422861743261288259915465, 6.03330214742327009150557463484, 6.50633388063324819338747364862, 7.14375287833410078479431736650, 7.78145398812000971262690871554
