| L(s) = 1 | − 3i·5-s + 5·7-s − 15i·11-s + 10·13-s − 18i·17-s + 16·19-s + 12i·23-s + 16·25-s + 30i·29-s − 31-s − 15i·35-s − 20·37-s − 60i·41-s − 50·43-s + 6i·47-s + ⋯ |
| L(s) = 1 | − 0.600i·5-s + 0.714·7-s − 1.36i·11-s + 0.769·13-s − 1.05i·17-s + 0.842·19-s + 0.521i·23-s + 0.640·25-s + 1.03i·29-s − 0.0322·31-s − 0.428i·35-s − 0.540·37-s − 1.46i·41-s − 1.16·43-s + 0.127i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.189233198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.189233198\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 3iT - 25T^{2} \) |
| 7 | \( 1 - 5T + 49T^{2} \) |
| 11 | \( 1 + 15iT - 121T^{2} \) |
| 13 | \( 1 - 10T + 169T^{2} \) |
| 17 | \( 1 + 18iT - 289T^{2} \) |
| 19 | \( 1 - 16T + 361T^{2} \) |
| 23 | \( 1 - 12iT - 529T^{2} \) |
| 29 | \( 1 - 30iT - 841T^{2} \) |
| 31 | \( 1 + T + 961T^{2} \) |
| 37 | \( 1 + 20T + 1.36e3T^{2} \) |
| 41 | \( 1 + 60iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 50T + 1.84e3T^{2} \) |
| 47 | \( 1 - 6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 27iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 30iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 76T + 3.72e3T^{2} \) |
| 67 | \( 1 - 10T + 4.48e3T^{2} \) |
| 71 | \( 1 - 90iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 65T + 5.32e3T^{2} \) |
| 79 | \( 1 - 14T + 6.24e3T^{2} \) |
| 83 | \( 1 - 3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 90iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 85T + 9.40e3T^{2} \) |
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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
−8.723659136660937307165516926271, −8.407941981301434943798293521686, −7.41324393877948669293309893521, −6.55832387415150657911825506639, −5.32134036081736609444724001651, −5.18092298823030786504609296435, −3.80433219835183863095069479350, −3.02339770472018527124582886577, −1.53875378303095680369120804449, −0.63281782683444973059285064470,
1.30353875071647683524153017130, 2.25581915237928960410422401550, 3.42754001122960623869534860313, 4.39170551437312454250036165073, 5.16596672103773687726846325395, 6.27681701977882467786375488291, 6.89549980845960599620069649713, 7.83979932901134235429165921120, 8.367915488285622921947510688527, 9.425988106809249620743146874220
