L(s) = 1 | − 2·7-s + 11-s + 4·13-s − 5·17-s − 19-s − 2·23-s + 8·29-s − 10·31-s − 6·37-s + 3·41-s − 4·43-s + 4·47-s − 3·49-s − 6·53-s + 8·59-s + 10·61-s + 67-s − 12·71-s + 3·73-s − 2·77-s − 6·79-s − 13·83-s + 9·89-s − 8·91-s − 14·97-s − 6·101-s − 4·103-s + ⋯ |
L(s) = 1 | − 0.755·7-s + 0.301·11-s + 1.10·13-s − 1.21·17-s − 0.229·19-s − 0.417·23-s + 1.48·29-s − 1.79·31-s − 0.986·37-s + 0.468·41-s − 0.609·43-s + 0.583·47-s − 3/7·49-s − 0.824·53-s + 1.04·59-s + 1.28·61-s + 0.122·67-s − 1.42·71-s + 0.351·73-s − 0.227·77-s − 0.675·79-s − 1.42·83-s + 0.953·89-s − 0.838·91-s − 1.42·97-s − 0.597·101-s − 0.394·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 13 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 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
−8.467326420441589290981592742883, −7.29467707966125468244974550807, −6.62017074939801966978952207822, −6.11381316189612319826601820106, −5.19567470588358634831197841025, −4.15733307169333325066118627913, −3.56777788939131847059582382296, −2.54938777815509649270854275721, −1.45185929724341557276486985203, 0,
1.45185929724341557276486985203, 2.54938777815509649270854275721, 3.56777788939131847059582382296, 4.15733307169333325066118627913, 5.19567470588358634831197841025, 6.11381316189612319826601820106, 6.62017074939801966978952207822, 7.29467707966125468244974550807, 8.467326420441589290981592742883