L(s) = 1 | + 15.6·5-s + (−3.87 + 18.1i)7-s + 39.5i·11-s + 64.1i·13-s − 56.6·17-s − 117. i·19-s + 7.61i·23-s + 119.·25-s + 45.9i·29-s + 290. i·31-s + (−60.6 + 283. i)35-s − 335.·37-s + 194.·41-s − 304.·43-s + 636.·47-s + ⋯ |
L(s) = 1 | + 1.39·5-s + (−0.209 + 0.977i)7-s + 1.08i·11-s + 1.36i·13-s − 0.808·17-s − 1.41i·19-s + 0.0690i·23-s + 0.959·25-s + 0.294i·29-s + 1.68i·31-s + (−0.293 + 1.36i)35-s − 1.49·37-s + 0.741·41-s − 1.08·43-s + 1.97·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.209i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.514181528\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.514181528\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (3.87 - 18.1i)T \) |
good | 5 | \( 1 - 15.6T + 125T^{2} \) |
| 11 | \( 1 - 39.5iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 64.1iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 56.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 117. iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 7.61iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 45.9iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 290. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 335.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 194.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 304.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 636.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 274. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 517.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 164. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 736.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 599. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 214. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 908.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 779.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 443.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 389. iT - 9.12e5T^{2} \) |
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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
−9.140348292267742884860920186304, −8.675979491766768965312293491032, −7.11664192869013917971645340078, −6.75330453587357196209692305891, −5.94662441121380899208588816288, −5.04614956005410667654575003115, −4.49831150909250941435918765272, −2.99055412022196330566020746328, −2.12300922351124397398226324156, −1.62469560692141885046712828894,
0.26871314066737299394917775027, 1.25190480508441058694520256034, 2.33048190701869072382021897894, 3.32393466858011966534870293583, 4.17105383506071514088047392039, 5.45612119124800813173908499440, 5.86864627663871227826177756103, 6.57685587199660788334974237032, 7.61337219925168582224731108604, 8.281923937665428985924387633722