| L(s) = 1 | − 2·7-s − 5·11-s − 13-s − 2·17-s − 8·19-s − 7·23-s + 4·29-s − 2·31-s + 9·37-s + 6·41-s + 8·43-s − 47-s − 3·49-s − 59-s + 7·61-s + 10·67-s + 3·71-s + 6·73-s + 10·77-s + 6·79-s + 16·83-s − 6·89-s + 2·91-s − 97-s + 6·101-s + 14·103-s − 9·107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1.50·11-s − 0.277·13-s − 0.485·17-s − 1.83·19-s − 1.45·23-s + 0.742·29-s − 0.359·31-s + 1.47·37-s + 0.937·41-s + 1.21·43-s − 0.145·47-s − 3/7·49-s − 0.130·59-s + 0.896·61-s + 1.22·67-s + 0.356·71-s + 0.702·73-s + 1.13·77-s + 0.675·79-s + 1.75·83-s − 0.635·89-s + 0.209·91-s − 0.101·97-s + 0.597·101-s + 1.37·103-s − 0.870·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8639188124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8639188124\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−8.033686795497423995167588485484, −7.66270619259160024518993845221, −6.52793918138760533330756832853, −6.19997513208859204273086348067, −5.29644105921340697484669724801, −4.44872690685319839960438338286, −3.77910760542096179234118422627, −2.55802295986724316491752633599, −2.24965982336126226949137693167, −0.46338223251070382936367119245,
0.46338223251070382936367119245, 2.24965982336126226949137693167, 2.55802295986724316491752633599, 3.77910760542096179234118422627, 4.44872690685319839960438338286, 5.29644105921340697484669724801, 6.19997513208859204273086348067, 6.52793918138760533330756832853, 7.66270619259160024518993845221, 8.033686795497423995167588485484
