L(s) = 1 | − 3-s − 5-s − 2·9-s + 3·11-s − 13-s + 15-s − 5·17-s − 6·19-s + 25-s + 5·27-s + 5·29-s − 2·31-s − 3·33-s + 4·37-s + 39-s − 2·41-s + 10·43-s + 2·45-s + 9·47-s + 5·51-s − 6·53-s − 3·55-s + 6·57-s − 6·59-s + 12·61-s + 65-s − 2·67-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 2/3·9-s + 0.904·11-s − 0.277·13-s + 0.258·15-s − 1.21·17-s − 1.37·19-s + 1/5·25-s + 0.962·27-s + 0.928·29-s − 0.359·31-s − 0.522·33-s + 0.657·37-s + 0.160·39-s − 0.312·41-s + 1.52·43-s + 0.298·45-s + 1.31·47-s + 0.700·51-s − 0.824·53-s − 0.404·55-s + 0.794·57-s − 0.781·59-s + 1.53·61-s + 0.124·65-s − 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−16.26915966367854, −15.79778185867017, −15.14321347797224, −14.62235626543621, −14.16441921773535, −13.51419893285825, −12.70204511330969, −12.39847613731656, −11.66698629651409, −11.29574618531519, −10.74637683497733, −10.25035192269419, −9.260419061274222, −8.839838640999551, −8.379929753374702, −7.532880943023111, −6.864245193616678, −6.261817671816554, −5.898779760669487, −4.856192615940822, −4.403737694432140, −3.769334643274700, −2.760046658424459, −2.112338427470954, −0.9087992454753121, 0,
0.9087992454753121, 2.112338427470954, 2.760046658424459, 3.769334643274700, 4.403737694432140, 4.856192615940822, 5.898779760669487, 6.261817671816554, 6.864245193616678, 7.532880943023111, 8.379929753374702, 8.839838640999551, 9.260419061274222, 10.25035192269419, 10.74637683497733, 11.29574618531519, 11.66698629651409, 12.39847613731656, 12.70204511330969, 13.51419893285825, 14.16441921773535, 14.62235626543621, 15.14321347797224, 15.79778185867017, 16.26915966367854