L(s) = 1 | − 3-s − 3·7-s + 9-s + 5·13-s − 5·19-s + 3·21-s + 4·23-s − 27-s − 4·29-s − 5·31-s + 10·37-s − 5·39-s − 10·41-s − 43-s − 2·47-s + 2·49-s + 10·53-s + 5·57-s + 10·59-s + 5·61-s − 3·63-s + 3·67-s − 4·69-s − 10·71-s − 10·73-s + 81-s + 14·83-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s + 1.38·13-s − 1.14·19-s + 0.654·21-s + 0.834·23-s − 0.192·27-s − 0.742·29-s − 0.898·31-s + 1.64·37-s − 0.800·39-s − 1.56·41-s − 0.152·43-s − 0.291·47-s + 2/7·49-s + 1.37·53-s + 0.662·57-s + 1.30·59-s + 0.640·61-s − 0.377·63-s + 0.366·67-s − 0.481·69-s − 1.18·71-s − 1.17·73-s + 1/9·81-s + 1.53·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−7.906874324032786268317512964023, −6.93782713135599512694488429612, −6.46200162108756387652083518288, −5.86909778457292125089701912577, −5.09670502966568039386523631453, −3.99559145448284582923133783989, −3.52856921465519946835638891404, −2.41869379380795584422549564850, −1.21084270882711870894759657959, 0,
1.21084270882711870894759657959, 2.41869379380795584422549564850, 3.52856921465519946835638891404, 3.99559145448284582923133783989, 5.09670502966568039386523631453, 5.86909778457292125089701912577, 6.46200162108756387652083518288, 6.93782713135599512694488429612, 7.906874324032786268317512964023