L(s) = 1 | − 5-s + 7-s − 3·9-s − 13-s − 2·17-s + 4·19-s + 25-s − 2·29-s − 35-s + 2·37-s + 6·41-s + 4·43-s + 3·45-s + 8·47-s + 49-s + 6·53-s + 4·59-s − 10·61-s − 3·63-s + 65-s − 12·67-s − 4·71-s − 10·73-s + 9·81-s − 12·83-s + 2·85-s − 18·89-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.377·7-s − 9-s − 0.277·13-s − 0.485·17-s + 0.917·19-s + 1/5·25-s − 0.371·29-s − 0.169·35-s + 0.328·37-s + 0.937·41-s + 0.609·43-s + 0.447·45-s + 1.16·47-s + 1/7·49-s + 0.824·53-s + 0.520·59-s − 1.28·61-s − 0.377·63-s + 0.124·65-s − 1.46·67-s − 0.474·71-s − 1.17·73-s + 81-s − 1.31·83-s + 0.216·85-s − 1.90·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.51130579904357732840659733736, −7.09919938407154638179485266362, −5.95757956225412542169589403935, −5.59587330588391857865009492155, −4.65658660100193562046579012240, −4.02721462820321817272857975691, −3.03539942710213026543664991451, −2.43327688196743013621038231227, −1.19570141677931663743820748832, 0,
1.19570141677931663743820748832, 2.43327688196743013621038231227, 3.03539942710213026543664991451, 4.02721462820321817272857975691, 4.65658660100193562046579012240, 5.59587330588391857865009492155, 5.95757956225412542169589403935, 7.09919938407154638179485266362, 7.51130579904357732840659733736