L(s) = 1 | + 3-s − 5-s + 2·7-s + 9-s − 5·11-s + 13-s − 15-s + 17-s − 19-s + 2·21-s − 4·23-s + 25-s + 27-s + 4·29-s − 3·31-s − 5·33-s − 2·35-s + 11·37-s + 39-s + 9·43-s − 45-s + 10·47-s − 3·49-s + 51-s + 2·53-s + 5·55-s − 57-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.50·11-s + 0.277·13-s − 0.258·15-s + 0.242·17-s − 0.229·19-s + 0.436·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s + 0.742·29-s − 0.538·31-s − 0.870·33-s − 0.338·35-s + 1.80·37-s + 0.160·39-s + 1.37·43-s − 0.149·45-s + 1.45·47-s − 3/7·49-s + 0.140·51-s + 0.274·53-s + 0.674·55-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.33613206325204, −12.77883759869063, −12.29505587320506, −11.97987588088093, −11.24123482794048, −10.85835391765557, −10.53657218576891, −10.02005808291549, −9.422725446403994, −8.915240530995935, −8.450088919544394, −7.865315945055728, −7.681085362660848, −7.405496910840245, −6.470533284390900, −5.999354886686278, −5.460025466036013, −4.893366993432879, −4.313441259363888, −4.050047975128889, −3.203917913961853, −2.678695777509468, −2.302569104766228, −1.529496126694154, −0.8418493261496333, 0,
0.8418493261496333, 1.529496126694154, 2.302569104766228, 2.678695777509468, 3.203917913961853, 4.050047975128889, 4.313441259363888, 4.893366993432879, 5.460025466036013, 5.999354886686278, 6.470533284390900, 7.405496910840245, 7.681085362660848, 7.865315945055728, 8.450088919544394, 8.915240530995935, 9.422725446403994, 10.02005808291549, 10.53657218576891, 10.85835391765557, 11.24123482794048, 11.97987588088093, 12.29505587320506, 12.77883759869063, 13.33613206325204