L(s) = 1 | + 5-s − 7-s − 3·9-s + 2·11-s − 3·17-s + 6·19-s − 4·23-s − 4·25-s + 7·29-s + 4·31-s − 35-s − 9·37-s + 9·41-s − 10·43-s − 3·45-s − 2·47-s + 49-s − 9·53-s + 2·55-s − 14·59-s + 5·61-s + 3·63-s + 8·67-s + 10·71-s − 7·73-s − 2·77-s + 2·79-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s − 9-s + 0.603·11-s − 0.727·17-s + 1.37·19-s − 0.834·23-s − 4/5·25-s + 1.29·29-s + 0.718·31-s − 0.169·35-s − 1.47·37-s + 1.40·41-s − 1.52·43-s − 0.447·45-s − 0.291·47-s + 1/7·49-s − 1.23·53-s + 0.269·55-s − 1.82·59-s + 0.640·61-s + 0.377·63-s + 0.977·67-s + 1.18·71-s − 0.819·73-s − 0.227·77-s + 0.225·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.22909648685045, −13.75116496102445, −13.60624749977999, −12.78845919640548, −12.19296345831926, −11.82181020552207, −11.43318138141217, −10.81599220686073, −10.18661417312130, −9.764884914192902, −9.253099394156371, −8.842613949819286, −8.131349975796546, −7.835894770230455, −6.975168111339528, −6.404001406521185, −6.165055548989551, −5.457261783224381, −4.939039738226868, −4.309488206199336, −3.445518696775843, −3.156342821290636, −2.346868827757274, −1.753794686686920, −0.8739813041508053, 0,
0.8739813041508053, 1.753794686686920, 2.346868827757274, 3.156342821290636, 3.445518696775843, 4.309488206199336, 4.939039738226868, 5.457261783224381, 6.165055548989551, 6.404001406521185, 6.975168111339528, 7.835894770230455, 8.131349975796546, 8.842613949819286, 9.253099394156371, 9.764884914192902, 10.18661417312130, 10.81599220686073, 11.43318138141217, 11.82181020552207, 12.19296345831926, 12.78845919640548, 13.60624749977999, 13.75116496102445, 14.22909648685045