| L(s) = 1 | + 3·3-s + 18i·5-s + 8i·7-s + 9·9-s + 36i·11-s + 54i·15-s − 18·17-s + 100i·19-s + 24i·21-s − 72·23-s − 199·25-s + 27·27-s − 234·29-s + 16i·31-s + 108i·33-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.60i·5-s + 0.431i·7-s + 0.333·9-s + 0.986i·11-s + 0.929i·15-s − 0.256·17-s + 1.20i·19-s + 0.249i·21-s − 0.652·23-s − 1.59·25-s + 0.192·27-s − 1.49·29-s + 0.0926i·31-s + 0.569i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.318143089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.318143089\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 18iT - 125T^{2} \) |
| 7 | \( 1 - 8iT - 343T^{2} \) |
| 11 | \( 1 - 36iT - 1.33e3T^{2} \) |
| 17 | \( 1 + 18T + 4.91e3T^{2} \) |
| 19 | \( 1 - 100iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 72T + 1.21e4T^{2} \) |
| 29 | \( 1 + 234T + 2.43e4T^{2} \) |
| 31 | \( 1 - 16iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 226iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 90iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 452T + 7.95e4T^{2} \) |
| 47 | \( 1 - 432iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 414T + 1.48e5T^{2} \) |
| 59 | \( 1 + 684iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 422T + 2.26e5T^{2} \) |
| 67 | \( 1 + 332iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 360iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 26iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 512T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.18e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 630iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.05e3iT - 9.12e5T^{2} \) |
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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
−9.440770602547108786361063509589, −8.345957964689191796166767793417, −7.58945692039053130876819925830, −7.02909610025355546473894881567, −6.24300115372090712571114381319, −5.38125840976665714624537795491, −4.03967053392608507458100283467, −3.45943586589687769923763457704, −2.37397160400869814343777185653, −1.86227310477759945181101407425,
0.24570329086269576981319465952, 1.08935316851399059537173668201, 2.15305536183307778070097210187, 3.42612834916423622981928327301, 4.21826435798735615537071362669, 5.02366455434393441947948880626, 5.76926548997243841302322716052, 6.87121693574072816399190878207, 7.74275349709763535703654400905, 8.612530794005377364556981245722
