L(s) = 1 | − 1.26·3-s − 0.339·5-s − 4.08·7-s − 1.39·9-s + 2.81·11-s − 3.28·13-s + 0.430·15-s − 3.05·17-s + 6.09·19-s + 5.17·21-s + 0.204·23-s − 4.88·25-s + 5.56·27-s − 2.58·29-s + 7.00·31-s − 3.56·33-s + 1.38·35-s + 2.47·37-s + 4.16·39-s + 4.44·41-s + 8.73·43-s + 0.472·45-s − 2.38·47-s + 9.67·49-s + 3.87·51-s + 1.43·53-s − 0.954·55-s + ⋯ |
L(s) = 1 | − 0.732·3-s − 0.151·5-s − 1.54·7-s − 0.463·9-s + 0.847·11-s − 0.910·13-s + 0.111·15-s − 0.740·17-s + 1.39·19-s + 1.12·21-s + 0.0425·23-s − 0.976·25-s + 1.07·27-s − 0.479·29-s + 1.25·31-s − 0.620·33-s + 0.234·35-s + 0.407·37-s + 0.666·39-s + 0.694·41-s + 1.33·43-s + 0.0704·45-s − 0.348·47-s + 1.38·49-s + 0.542·51-s + 0.196·53-s − 0.128·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7521820019\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7521820019\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 167 | \( 1 - T \) |
good | 3 | \( 1 + 1.26T + 3T^{2} \) |
| 5 | \( 1 + 0.339T + 5T^{2} \) |
| 7 | \( 1 + 4.08T + 7T^{2} \) |
| 11 | \( 1 - 2.81T + 11T^{2} \) |
| 13 | \( 1 + 3.28T + 13T^{2} \) |
| 17 | \( 1 + 3.05T + 17T^{2} \) |
| 19 | \( 1 - 6.09T + 19T^{2} \) |
| 23 | \( 1 - 0.204T + 23T^{2} \) |
| 29 | \( 1 + 2.58T + 29T^{2} \) |
| 31 | \( 1 - 7.00T + 31T^{2} \) |
| 37 | \( 1 - 2.47T + 37T^{2} \) |
| 41 | \( 1 - 4.44T + 41T^{2} \) |
| 43 | \( 1 - 8.73T + 43T^{2} \) |
| 47 | \( 1 + 2.38T + 47T^{2} \) |
| 53 | \( 1 - 1.43T + 53T^{2} \) |
| 59 | \( 1 - 5.48T + 59T^{2} \) |
| 61 | \( 1 + 5.24T + 61T^{2} \) |
| 67 | \( 1 + 9.65T + 67T^{2} \) |
| 71 | \( 1 + 1.65T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 - 6.14T + 79T^{2} \) |
| 83 | \( 1 - 1.56T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 - 2.47T + 97T^{2} \) |
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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
−9.499065121970245825467000444204, −9.182235642739005870652929026992, −7.86591397493081278363450346158, −6.97238672596746600875212803764, −6.27955920371159052478084984410, −5.64562593628741481572171567579, −4.53967827121825141249346459736, −3.48857561930286863394113977217, −2.54869721513187973697415170072, −0.62905087182049633332248604445,
0.62905087182049633332248604445, 2.54869721513187973697415170072, 3.48857561930286863394113977217, 4.53967827121825141249346459736, 5.64562593628741481572171567579, 6.27955920371159052478084984410, 6.97238672596746600875212803764, 7.86591397493081278363450346158, 9.182235642739005870652929026992, 9.499065121970245825467000444204