L(s) = 1 | + 0.925·3-s − 5-s − 0.763·7-s − 2.14·9-s + 6.44·11-s − 2.67·13-s − 0.925·15-s + 5.30·17-s − 4.88·19-s − 0.706·21-s − 0.149·23-s + 25-s − 4.75·27-s + 5.21·29-s − 0.912·31-s + 5.96·33-s + 0.763·35-s + 37-s − 2.47·39-s + 10.7·41-s + 8.12·43-s + 2.14·45-s − 3.51·47-s − 6.41·49-s + 4.90·51-s + 10.5·53-s − 6.44·55-s + ⋯ |
L(s) = 1 | + 0.534·3-s − 0.447·5-s − 0.288·7-s − 0.714·9-s + 1.94·11-s − 0.740·13-s − 0.238·15-s + 1.28·17-s − 1.12·19-s − 0.154·21-s − 0.0312·23-s + 0.200·25-s − 0.915·27-s + 0.968·29-s − 0.163·31-s + 1.03·33-s + 0.129·35-s + 0.164·37-s − 0.395·39-s + 1.68·41-s + 1.23·43-s + 0.319·45-s − 0.511·47-s − 0.916·49-s + 0.686·51-s + 1.45·53-s − 0.869·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.981666360\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.981666360\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 - T \) |
good | 3 | \( 1 - 0.925T + 3T^{2} \) |
| 7 | \( 1 + 0.763T + 7T^{2} \) |
| 11 | \( 1 - 6.44T + 11T^{2} \) |
| 13 | \( 1 + 2.67T + 13T^{2} \) |
| 17 | \( 1 - 5.30T + 17T^{2} \) |
| 19 | \( 1 + 4.88T + 19T^{2} \) |
| 23 | \( 1 + 0.149T + 23T^{2} \) |
| 29 | \( 1 - 5.21T + 29T^{2} \) |
| 31 | \( 1 + 0.912T + 31T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 - 8.12T + 43T^{2} \) |
| 47 | \( 1 + 3.51T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 + 4.45T + 59T^{2} \) |
| 61 | \( 1 + 8.26T + 61T^{2} \) |
| 67 | \( 1 - 6.88T + 67T^{2} \) |
| 71 | \( 1 + 2.03T + 71T^{2} \) |
| 73 | \( 1 - 9.81T + 73T^{2} \) |
| 79 | \( 1 - 1.83T + 79T^{2} \) |
| 83 | \( 1 + 1.03T + 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 + 5.18T + 97T^{2} \) |
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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
−8.827127692710184844062684408548, −8.027362989072120879831714240246, −7.35286813041725819739773062643, −6.43536367349372381955790179986, −5.88367777555403592038435916499, −4.66868031112719792972498436426, −3.88891899918750285583614688970, −3.18938689142215102552038326759, −2.19343231803019547596089387265, −0.848165116974752923147731731111,
0.848165116974752923147731731111, 2.19343231803019547596089387265, 3.18938689142215102552038326759, 3.88891899918750285583614688970, 4.66868031112719792972498436426, 5.88367777555403592038435916499, 6.43536367349372381955790179986, 7.35286813041725819739773062643, 8.027362989072120879831714240246, 8.827127692710184844062684408548