L(s) = 1 | + 0.340·2-s − 1.21·3-s − 1.88·4-s + 3.18·5-s − 0.413·6-s − 4.05·7-s − 1.32·8-s − 1.52·9-s + 1.08·10-s − 11-s + 2.28·12-s − 1.38·14-s − 3.86·15-s + 3.31·16-s − 2.34·17-s − 0.520·18-s + 6.23·19-s − 6.00·20-s + 4.91·21-s − 0.340·22-s − 8.20·23-s + 1.60·24-s + 5.16·25-s + 5.49·27-s + 7.63·28-s − 4.01·29-s − 1.31·30-s + ⋯ |
L(s) = 1 | + 0.240·2-s − 0.700·3-s − 0.942·4-s + 1.42·5-s − 0.168·6-s − 1.53·7-s − 0.467·8-s − 0.509·9-s + 0.343·10-s − 0.301·11-s + 0.659·12-s − 0.368·14-s − 0.998·15-s + 0.829·16-s − 0.569·17-s − 0.122·18-s + 1.43·19-s − 1.34·20-s + 1.07·21-s − 0.0725·22-s − 1.71·23-s + 0.327·24-s + 1.03·25-s + 1.05·27-s + 1.44·28-s − 0.745·29-s − 0.240·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9234585173\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9234585173\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 0.340T + 2T^{2} \) |
| 3 | \( 1 + 1.21T + 3T^{2} \) |
| 5 | \( 1 - 3.18T + 5T^{2} \) |
| 7 | \( 1 + 4.05T + 7T^{2} \) |
| 17 | \( 1 + 2.34T + 17T^{2} \) |
| 19 | \( 1 - 6.23T + 19T^{2} \) |
| 23 | \( 1 + 8.20T + 23T^{2} \) |
| 29 | \( 1 + 4.01T + 29T^{2} \) |
| 31 | \( 1 + 0.723T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 - 4.40T + 41T^{2} \) |
| 43 | \( 1 - 9.58T + 43T^{2} \) |
| 47 | \( 1 + 4.76T + 47T^{2} \) |
| 53 | \( 1 - 5.18T + 53T^{2} \) |
| 59 | \( 1 - 0.694T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 7.17T + 67T^{2} \) |
| 71 | \( 1 + 2.32T + 71T^{2} \) |
| 73 | \( 1 - 2.44T + 73T^{2} \) |
| 79 | \( 1 + 2.59T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−9.549782685367901871577400733513, −8.755410559427730801768400825775, −7.59473930198934904877598300964, −6.42131241336286482981536985544, −5.75678281161993433365647296376, −5.64153909396892585190587547292, −4.41420243576565155637967719961, −3.32418988874842526282720735266, −2.39579744214355782164279208365, −0.62977272182608418160539578130,
0.62977272182608418160539578130, 2.39579744214355782164279208365, 3.32418988874842526282720735266, 4.41420243576565155637967719961, 5.64153909396892585190587547292, 5.75678281161993433365647296376, 6.42131241336286482981536985544, 7.59473930198934904877598300964, 8.755410559427730801768400825775, 9.549782685367901871577400733513