L(s) = 1 | − 5-s − 3·11-s − 13-s + 3·17-s + 23-s + 25-s − 6·29-s + 10·31-s − 9·37-s + 3·41-s − 10·43-s + 2·47-s − 10·53-s + 3·55-s − 9·59-s + 2·61-s + 65-s − 5·67-s + 7·73-s − 79-s + 10·83-s − 3·85-s − 3·89-s − 7·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.904·11-s − 0.277·13-s + 0.727·17-s + 0.208·23-s + 1/5·25-s − 1.11·29-s + 1.79·31-s − 1.47·37-s + 0.468·41-s − 1.52·43-s + 0.291·47-s − 1.37·53-s + 0.404·55-s − 1.17·59-s + 0.256·61-s + 0.124·65-s − 0.610·67-s + 0.819·73-s − 0.112·79-s + 1.09·83-s − 0.325·85-s − 0.317·89-s − 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 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
−13.19290708166357, −12.58625501837335, −12.25356750509529, −11.85647521246571, −11.31506460148921, −10.77875962312959, −10.46574507437951, −9.864273276796783, −9.551280755861637, −8.912620841622720, −8.312153178218013, −8.004009907039965, −7.574382788241299, −7.019634144735394, −6.566942203054436, −5.906505462604163, −5.439937877043795, −4.848119188828490, −4.577366746138558, −3.768786088139613, −3.211169180564492, −2.900732060319479, −2.080218027698897, −1.528505659668996, −0.6772048347992868, 0,
0.6772048347992868, 1.528505659668996, 2.080218027698897, 2.900732060319479, 3.211169180564492, 3.768786088139613, 4.577366746138558, 4.848119188828490, 5.439937877043795, 5.906505462604163, 6.566942203054436, 7.019634144735394, 7.574382788241299, 8.004009907039965, 8.312153178218013, 8.912620841622720, 9.551280755861637, 9.864273276796783, 10.46574507437951, 10.77875962312959, 11.31506460148921, 11.85647521246571, 12.25356750509529, 12.58625501837335, 13.19290708166357