| L(s) = 1 | + i·2-s + 7·4-s + 24i·7-s + 15i·8-s − 52·11-s + 22i·13-s − 24·14-s + 41·16-s − 14i·17-s + 20·19-s − 52i·22-s + 168i·23-s − 22·26-s + 168i·28-s + 230·29-s + ⋯ |
| L(s) = 1 | + 0.353i·2-s + 0.875·4-s + 1.29i·7-s + 0.662i·8-s − 1.42·11-s + 0.469i·13-s − 0.458·14-s + 0.640·16-s − 0.199i·17-s + 0.241·19-s − 0.503i·22-s + 1.52i·23-s − 0.165·26-s + 1.13i·28-s + 1.47·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.928021 + 1.50157i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.928021 + 1.50157i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - iT - 8T^{2} \) |
| 7 | \( 1 - 24iT - 343T^{2} \) |
| 11 | \( 1 + 52T + 1.33e3T^{2} \) |
| 13 | \( 1 - 22iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 14iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 20T + 6.85e3T^{2} \) |
| 23 | \( 1 - 168iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 230T + 2.43e4T^{2} \) |
| 31 | \( 1 + 288T + 2.97e4T^{2} \) |
| 37 | \( 1 - 34iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 122T + 6.89e4T^{2} \) |
| 43 | \( 1 + 188iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 256iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 338iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 100T + 2.05e5T^{2} \) |
| 61 | \( 1 - 742T + 2.26e5T^{2} \) |
| 67 | \( 1 - 84iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 328T + 3.57e5T^{2} \) |
| 73 | \( 1 + 38iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 240T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.21e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 330T + 7.04e5T^{2} \) |
| 97 | \( 1 + 866iT - 9.12e5T^{2} \) |
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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
−11.99958183134376279732849961531, −11.28883901914629846712481762418, −10.24284093269802154477966409202, −9.048411507020811325532386508582, −8.016594154601279002216180292585, −7.11357940921124990242723898276, −5.84012799422825265211009764263, −5.18088861471269145153166646197, −3.05980952932106609294136929095, −2.01294238944729982403051570765,
0.68620481326327830775892603488, 2.40800369120816277172154862329, 3.64713644689738041200800205725, 5.10895126975731395221963346757, 6.53740173583214223133516757163, 7.42422128237927050165591689644, 8.294172957580566884743473764199, 10.10114200517281619199281432569, 10.47473042842716163997457062392, 11.25571480624891189520173440614
