L(s) = 1 | + (−2.52 + 4.54i)3-s + 8.01i·5-s − 12.6i·7-s + (−14.2 − 22.9i)9-s + 37.7·11-s + 60.0·13-s + (−36.3 − 20.2i)15-s + 37.0i·17-s − 127. i·19-s + (57.2 + 31.7i)21-s + 56.9·23-s + 60.8·25-s + (140. − 7.09i)27-s + 220. i·29-s + 2.26i·31-s + ⋯ |
L(s) = 1 | + (−0.485 + 0.874i)3-s + 0.716i·5-s − 0.680i·7-s + (−0.528 − 0.848i)9-s + 1.03·11-s + 1.28·13-s + (−0.626 − 0.347i)15-s + 0.528i·17-s − 1.54i·19-s + (0.594 + 0.330i)21-s + 0.516·23-s + 0.486·25-s + (0.998 − 0.0505i)27-s + 1.41i·29-s + 0.0130i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.732058327\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.732058327\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.52 - 4.54i)T \) |
good | 5 | \( 1 - 8.01iT - 125T^{2} \) |
| 7 | \( 1 + 12.6iT - 343T^{2} \) |
| 11 | \( 1 - 37.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 60.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 37.0iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 127. iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 56.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 220. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 2.26iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 166.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 154. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 53.4iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 591.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 538. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 586.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 431.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 175. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 29.5T + 3.57e5T^{2} \) |
| 73 | \( 1 - 937.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 409. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 1.29e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 921. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.64e3T + 9.12e5T^{2} \) |
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
−10.97359737509117853226377080356, −10.43189789729993105152921265422, −9.273333739710059501658478118044, −8.579369283913083002552204693117, −6.90221902167751377964381901427, −6.48337318560417944179169892679, −5.12643679720468462023984708967, −3.98926542544196161210251043093, −3.18879923532995388802470459273, −1.03220364771568435943649299204,
0.860753625846657256601063607050, 1.92326852211149962244417737984, 3.66835932129934732175665347017, 5.11637448516900733009659933828, 6.00463653082656915243775992543, 6.78157410047190405317096774524, 8.150592877678258226978012298790, 8.671515118572472253641982288354, 9.758854535236352177370778193713, 11.07953102363579660466825495375