| L(s) = 1 | − 2.73i·7-s − 2i·11-s − 3.46·13-s − 3.46i·17-s + 7.46i·19-s + 4.19i·23-s − 6.92i·29-s − 1.46·31-s − 2·37-s + 5.46·41-s − 8.73·43-s − 6.73i·47-s − 0.464·49-s + 4.53·53-s + 0.535i·59-s + ⋯ |
| L(s) = 1 | − 1.03i·7-s − 0.603i·11-s − 0.960·13-s − 0.840i·17-s + 1.71i·19-s + 0.874i·23-s − 1.28i·29-s − 0.262·31-s − 0.328·37-s + 0.853·41-s − 1.33·43-s − 0.981i·47-s − 0.0663·49-s + 0.623·53-s + 0.0697i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.663 - 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.663 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1725468691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1725468691\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2.73iT - 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 - 7.46iT - 19T^{2} \) |
| 23 | \( 1 - 4.19iT - 23T^{2} \) |
| 29 | \( 1 + 6.92iT - 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 5.46T + 41T^{2} \) |
| 43 | \( 1 + 8.73T + 43T^{2} \) |
| 47 | \( 1 + 6.73iT - 47T^{2} \) |
| 53 | \( 1 - 4.53T + 53T^{2} \) |
| 59 | \( 1 - 0.535iT - 59T^{2} \) |
| 61 | \( 1 - 4.92iT - 61T^{2} \) |
| 67 | \( 1 + 7.26T + 67T^{2} \) |
| 71 | \( 1 + 1.46T + 71T^{2} \) |
| 73 | \( 1 - 0.535iT - 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 + 4.73T + 83T^{2} \) |
| 89 | \( 1 + 4.92T + 89T^{2} \) |
| 97 | \( 1 + 6.39iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.037216236601841589431914515779, −7.43016210077906300315057850847, −7.00734241895843879182742540564, −5.97620635819328498619916111911, −5.46438549289637725168692305558, −4.52473949022915321212562589062, −3.85657701145369572446176919058, −3.15018119073131721363088142131, −2.10969654315944446021068028632, −1.06693829084178218461984811470,
0.04327277856563825331610319856, 1.57580144482513120865169523724, 2.48329918290764394064065553457, 3.00203764361140443480037662062, 4.23247381563558466752547545755, 4.88104673854746332168811270630, 5.45062912621268445484275016266, 6.34025940308613026284914044212, 6.98344959303434301360657313848, 7.56634726952813264054766523333
