L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 5·11-s − 12-s − 5·13-s + 16-s − 4·17-s − 18-s + 7·19-s + 5·22-s − 23-s + 24-s + 5·26-s − 27-s + 2·31-s − 32-s + 5·33-s + 4·34-s + 36-s − 37-s − 7·38-s + 5·39-s − 5·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.50·11-s − 0.288·12-s − 1.38·13-s + 1/4·16-s − 0.970·17-s − 0.235·18-s + 1.60·19-s + 1.06·22-s − 0.208·23-s + 0.204·24-s + 0.980·26-s − 0.192·27-s + 0.359·31-s − 0.176·32-s + 0.870·33-s + 0.685·34-s + 1/6·36-s − 0.164·37-s − 1.13·38-s + 0.800·39-s − 0.780·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4321388969\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4321388969\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.985633955569126326104729876451, −7.09086094597183228770957734691, −6.87323881498780805092396324266, −5.72749018395908600875471651194, −5.14504081518621556110382025569, −4.64202055187933863844772306464, −3.30436893004189723165100166651, −2.58104892053336577195611885354, −1.70828511571778319504603473078, −0.36598561046433104144469437791,
0.36598561046433104144469437791, 1.70828511571778319504603473078, 2.58104892053336577195611885354, 3.30436893004189723165100166651, 4.64202055187933863844772306464, 5.14504081518621556110382025569, 5.72749018395908600875471651194, 6.87323881498780805092396324266, 7.09086094597183228770957734691, 7.985633955569126326104729876451