L(s) = 1 | + 3-s + 9-s − 2·11-s + 4·13-s − 6·17-s − 8·19-s − 6·23-s + 27-s − 10·29-s − 4·31-s − 2·33-s − 6·37-s + 4·39-s − 6·41-s + 4·43-s + 8·47-s − 6·51-s − 2·53-s − 8·57-s + 4·59-s − 8·61-s − 8·67-s − 6·69-s + 10·71-s − 4·73-s − 4·79-s + 81-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.603·11-s + 1.10·13-s − 1.45·17-s − 1.83·19-s − 1.25·23-s + 0.192·27-s − 1.85·29-s − 0.718·31-s − 0.348·33-s − 0.986·37-s + 0.640·39-s − 0.937·41-s + 0.609·43-s + 1.16·47-s − 0.840·51-s − 0.274·53-s − 1.05·57-s + 0.520·59-s − 1.02·61-s − 0.977·67-s − 0.722·69-s + 1.18·71-s − 0.468·73-s − 0.450·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9350458600\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9350458600\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−14.28389376589020, −13.73704758600589, −13.34580316240201, −12.94605152268320, −12.47839627831559, −11.82578801114932, −11.03711383085391, −10.78779513795706, −10.43175734732209, −9.583894084312301, −9.082113731458612, −8.606637853128671, −8.267822588155212, −7.619463783458375, −6.998105576579175, −6.469272833569611, −5.878952334575741, −5.371813297728573, −4.407141042248691, −4.067737247454236, −3.561729205124095, −2.689602807900176, −1.933214549194850, −1.760252643881829, −0.2917964461293871,
0.2917964461293871, 1.760252643881829, 1.933214549194850, 2.689602807900176, 3.561729205124095, 4.067737247454236, 4.407141042248691, 5.371813297728573, 5.878952334575741, 6.469272833569611, 6.998105576579175, 7.619463783458375, 8.267822588155212, 8.606637853128671, 9.082113731458612, 9.583894084312301, 10.43175734732209, 10.78779513795706, 11.03711383085391, 11.82578801114932, 12.47839627831559, 12.94605152268320, 13.34580316240201, 13.73704758600589, 14.28389376589020