| L(s) = 1 | + (1 + i)3-s + (−19 + 19i)7-s − 79i·9-s − 202·11-s + (99 + 99i)13-s + (239 − 239i)17-s + 40i·19-s − 38·21-s + (541 + 541i)23-s + (160 − 160i)27-s + 200i·29-s + 758·31-s + (−202 − 202i)33-s + (−141 + 141i)37-s + 198i·39-s + ⋯ |
| L(s) = 1 | + (0.111 + 0.111i)3-s + (−0.387 + 0.387i)7-s − 0.975i·9-s − 1.66·11-s + (0.585 + 0.585i)13-s + (0.826 − 0.826i)17-s + 0.110i·19-s − 0.0861·21-s + (1.02 + 1.02i)23-s + (0.219 − 0.219i)27-s + 0.237i·29-s + 0.788·31-s + (−0.185 − 0.185i)33-s + (−0.102 + 0.102i)37-s + 0.130i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.737262896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.737262896\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (-1 - i)T + 81iT^{2} \) |
| 7 | \( 1 + (19 - 19i)T - 2.40e3iT^{2} \) |
| 11 | \( 1 + 202T + 1.46e4T^{2} \) |
| 13 | \( 1 + (-99 - 99i)T + 2.85e4iT^{2} \) |
| 17 | \( 1 + (-239 + 239i)T - 8.35e4iT^{2} \) |
| 19 | \( 1 - 40iT - 1.30e5T^{2} \) |
| 23 | \( 1 + (-541 - 541i)T + 2.79e5iT^{2} \) |
| 29 | \( 1 - 200iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 758T + 9.23e5T^{2} \) |
| 37 | \( 1 + (141 - 141i)T - 1.87e6iT^{2} \) |
| 41 | \( 1 - 1.04e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + (759 + 759i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + (459 - 459i)T - 4.87e6iT^{2} \) |
| 53 | \( 1 + (-1.81e3 - 1.81e3i)T + 7.89e6iT^{2} \) |
| 59 | \( 1 - 4.60e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 2.08e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + (-5.08e3 + 5.08e3i)T - 2.01e7iT^{2} \) |
| 71 | \( 1 - 3.47e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (-3.47e3 - 3.47e3i)T + 2.83e7iT^{2} \) |
| 79 | \( 1 + 7.68e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (-6.08e3 - 6.08e3i)T + 4.74e7iT^{2} \) |
| 89 | \( 1 - 5.68e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (561 - 561i)T - 8.85e7iT^{2} \) |
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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
−10.68689138663013968532059969094, −9.708005266849248212602686325765, −9.051672661225350123619609264332, −7.970514844440842449184831378330, −6.99597750571766061785841767432, −5.89096410447756101411900770840, −4.99533928209570691286239003331, −3.53151697540289938751978429173, −2.64987863341247488337536180758, −0.886544906324745306484897915128,
0.64663781710665956281077495271, 2.31091479658257957665059452829, 3.35580476666520696700716649448, 4.82073919989512673996079293970, 5.65813194700031470701462559488, 6.89599229084264690259156278567, 7.995502519492353190216280757955, 8.377419750323141452871688536595, 9.979919091010681488928090983602, 10.49158500194937339125995954296
