L(s) = 1 | − 2·5-s + 7-s − 3·9-s + 4·13-s + 6·17-s − 19-s + 4·23-s − 25-s − 6·29-s − 4·31-s − 2·35-s + 2·37-s + 4·43-s + 6·45-s + 8·47-s + 49-s − 2·53-s − 12·59-s − 2·61-s − 3·63-s − 8·65-s − 2·67-s + 14·71-s − 10·73-s + 10·79-s + 9·81-s − 12·83-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 0.377·7-s − 9-s + 1.10·13-s + 1.45·17-s − 0.229·19-s + 0.834·23-s − 1/5·25-s − 1.11·29-s − 0.718·31-s − 0.338·35-s + 0.328·37-s + 0.609·43-s + 0.894·45-s + 1.16·47-s + 1/7·49-s − 0.274·53-s − 1.56·59-s − 0.256·61-s − 0.377·63-s − 0.992·65-s − 0.244·67-s + 1.66·71-s − 1.17·73-s + 1.12·79-s + 81-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.554057354\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.554057354\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−7.67928801906448215480762876866, −7.43128082610777157045301112135, −6.26357311987761536797245991789, −5.72391554274944483418877307361, −5.09456147234369078039207519917, −4.07253590685321109288160537140, −3.54005313019569073249021460623, −2.84605625796708887845246644139, −1.65179600894541683516009280436, −0.62393739310250661812509860261,
0.62393739310250661812509860261, 1.65179600894541683516009280436, 2.84605625796708887845246644139, 3.54005313019569073249021460623, 4.07253590685321109288160537140, 5.09456147234369078039207519917, 5.72391554274944483418877307361, 6.26357311987761536797245991789, 7.43128082610777157045301112135, 7.67928801906448215480762876866