L(s) = 1 | + 2.36·2-s − 3·3-s − 2.42·4-s + 6.42·5-s − 7.08·6-s − 29.4·7-s − 24.6·8-s + 9·9-s + 15.1·10-s − 0.624·11-s + 7.26·12-s − 69.6·14-s − 19.2·15-s − 38.7·16-s + 87.7·17-s + 21.2·18-s + 82.8·19-s − 15.5·20-s + 88.4·21-s − 1.47·22-s − 74.7·23-s + 73.8·24-s − 83.7·25-s − 27·27-s + 71.4·28-s + 226.·29-s − 45.5·30-s + ⋯ |
L(s) = 1 | + 0.835·2-s − 0.577·3-s − 0.302·4-s + 0.574·5-s − 0.482·6-s − 1.59·7-s − 1.08·8-s + 0.333·9-s + 0.479·10-s − 0.0171·11-s + 0.174·12-s − 1.32·14-s − 0.331·15-s − 0.605·16-s + 1.25·17-s + 0.278·18-s + 0.999·19-s − 0.173·20-s + 0.919·21-s − 0.0142·22-s − 0.678·23-s + 0.628·24-s − 0.670·25-s − 0.192·27-s + 0.482·28-s + 1.44·29-s − 0.276·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.685016938\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.685016938\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 2.36T + 8T^{2} \) |
| 5 | \( 1 - 6.42T + 125T^{2} \) |
| 7 | \( 1 + 29.4T + 343T^{2} \) |
| 11 | \( 1 + 0.624T + 1.33e3T^{2} \) |
| 17 | \( 1 - 87.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 82.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 74.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 226.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 173.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 112.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 267.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 383.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 337.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 146.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 529.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 203.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 121.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 661.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 167.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 101.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 506.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.40e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.90e3T + 9.12e5T^{2} \) |
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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
−10.14995921365836295370013161200, −9.902550313303509560237305585432, −8.977523093989725465056687339119, −7.57098520382177320938106994553, −6.22728171331706402681004583156, −5.97755474265149850402812630461, −4.90561678015233118871595437802, −3.70287919248222052545906968165, −2.82373323396071499020193185827, −0.72386101712323911053632574865,
0.72386101712323911053632574865, 2.82373323396071499020193185827, 3.70287919248222052545906968165, 4.90561678015233118871595437802, 5.97755474265149850402812630461, 6.22728171331706402681004583156, 7.57098520382177320938106994553, 8.977523093989725465056687339119, 9.902550313303509560237305585432, 10.14995921365836295370013161200