L(s) = 1 | + 3·3-s + 7-s + 6·9-s + 11-s + 7·13-s − 3·17-s + 8·19-s + 3·21-s − 23-s + 9·27-s + 5·29-s − 2·31-s + 3·33-s − 4·37-s + 21·39-s − 8·41-s + 6·43-s − 3·47-s + 49-s − 9·51-s + 2·53-s + 24·57-s − 2·59-s + 14·61-s + 6·63-s − 4·67-s − 3·69-s + ⋯ |
L(s) = 1 | + 1.73·3-s + 0.377·7-s + 2·9-s + 0.301·11-s + 1.94·13-s − 0.727·17-s + 1.83·19-s + 0.654·21-s − 0.208·23-s + 1.73·27-s + 0.928·29-s − 0.359·31-s + 0.522·33-s − 0.657·37-s + 3.36·39-s − 1.24·41-s + 0.914·43-s − 0.437·47-s + 1/7·49-s − 1.26·51-s + 0.274·53-s + 3.17·57-s − 0.260·59-s + 1.79·61-s + 0.755·63-s − 0.488·67-s − 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(9.286765766\) |
\(L(\frac12)\) |
\(\approx\) |
\(9.286765766\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−13.19835647104445, −12.42733480396244, −11.92559890664094, −11.40024182054760, −11.01876571880876, −10.35903652040115, −10.00256910563931, −9.356304770571324, −9.038474246805575, −8.614729733473750, −8.259338016183114, −7.819004689915662, −7.336050023750309, −6.681981605480228, −6.442148981422151, −5.557421580781209, −5.131677795198493, −4.409099550528736, −3.820382596999591, −3.559171796939922, −3.072929801361612, −2.436120441433918, −1.807266157017956, −1.315607198277527, −0.7765734662465899,
0.7765734662465899, 1.315607198277527, 1.807266157017956, 2.436120441433918, 3.072929801361612, 3.559171796939922, 3.820382596999591, 4.409099550528736, 5.131677795198493, 5.557421580781209, 6.442148981422151, 6.681981605480228, 7.336050023750309, 7.819004689915662, 8.259338016183114, 8.614729733473750, 9.038474246805575, 9.356304770571324, 10.00256910563931, 10.35903652040115, 11.01876571880876, 11.40024182054760, 11.92559890664094, 12.42733480396244, 13.19835647104445